# Maple integration test file: "7 Inverse hyperbolic functions\7.1 Inverse hyperbolic sine\7.1.5 Inverse hyperbolic sine functions.txt"

lst:=[

# Integrands of the form u (a+b ArcSinh[c x])^n

# Integrands of the form (d+e x)^m (a+b ArcSinh[c x])^n

# Integrands of the form (d+e x)^m ArcSinh[c x]^n

# n>0
[arcsinh(c*x)/(d+e*x),x,8,-1/2*arcsinh(c*x)^2/e+arcsinh(c*x)*log(1+e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/e+arcsinh(c*x)*log(1+e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/e+polylog(2,-e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/e+polylog(2,-e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/e],
[arcsinh(c*x)^2/(d+e*x),x,10,-1/3*arcsinh(c*x)^3/e+arcsinh(c*x)^2*log(1+e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/e+arcsinh(c*x)^2*log(1+e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/e+2*arcsinh(c*x)*polylog(2,-e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/e+2*arcsinh(c*x)*polylog(2,-e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/e-2*polylog(3,-e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/e-2*polylog(3,-e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/e],
[arcsinh(c*x)^3/(d+e*x),x,12,-1/4*arcsinh(c*x)^4/e+arcsinh(c*x)^3*log(1+e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/e+arcsinh(c*x)^3*log(1+e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/e+3*arcsinh(c*x)^2*polylog(2,-e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/e+3*arcsinh(c*x)^2*polylog(2,-e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/e-6*arcsinh(c*x)*polylog(3,-e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/e-6*arcsinh(c*x)*polylog(3,-e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/e+6*polylog(4,-e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/e+6*polylog(4,-e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/e],

# n<0

# Integrands of the form (d+e x)^m (a+b ArcSinh[c x])^n

# n>0
[(d+e*x)^3*(a+b*arcsinh(c*x)),x,5,-1/32*b*(8*c^4*d^4-24*c^2*d^2*e^2+3*e^4)*arcsinh(c*x)/(c^4*e)+1/4*(d+e*x)^4*(a+b*arcsinh(c*x))/e-7/48*b*d*(d+e*x)^2*sqrt(1+c^2*x^2)/c-1/16*b*(d+e*x)^3*sqrt(1+c^2*x^2)/c-1/96*b*(4*d*(19*c^2*d^2-16*e^2)+e*(26*c^2*d^2-9*e^2)*x)*sqrt(1+c^2*x^2)/c^3],
[(d+e*x)^2*(a+b*arcsinh(c*x)),x,4,-1/6*b*d*(2*d^2-3*e^2/c^2)*arcsinh(c*x)/e+1/3*(d+e*x)^3*(a+b*arcsinh(c*x))/e-1/9*b*(d+e*x)^2*sqrt(1+c^2*x^2)/c-1/18*b*(4*(4*c^2*d^2-e^2)+5*c^2*d*e*x)*sqrt(1+c^2*x^2)/c^3],
[(d+e*x)*(a+b*arcsinh(c*x)),x,4,-1/4*b*(2*d^2-e^2/c^2)*arcsinh(c*x)/e+1/2*(d+e*x)^2*(a+b*arcsinh(c*x))/e-3/4*b*d*sqrt(1+c^2*x^2)/c-1/4*b*(d+e*x)*sqrt(1+c^2*x^2)/c],
[a+b*arcsinh(c*x),x,3,a*x+b*x*arcsinh(c*x)-b*sqrt(1+c^2*x^2)/c],
[(a+b*arcsinh(c*x))/(d+e*x),x,8,-1/2*(a+b*arcsinh(c*x))^2/(b*e)+(a+b*arcsinh(c*x))*log(1+e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/e+(a+b*arcsinh(c*x))*log(1+e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/e+b*polylog(2,-e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/e+b*polylog(2,-e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/e],
[(a+b*arcsinh(c*x))/(d+e*x)^2,x,3,(-a-b*arcsinh(c*x))/(e*(d+e*x))-b*c*arctanh((e-c^2*d*x)/(sqrt(c^2*d^2+e^2)*sqrt(1+c^2*x^2)))/(e*sqrt(c^2*d^2+e^2))],
[(a+b*arcsinh(c*x))/(d+e*x)^3,x,4,1/2*(-a-b*arcsinh(c*x))/(e*(d+e*x)^2)-1/2*b*c^3*d*arctanh((e-c^2*d*x)/(sqrt(c^2*d^2+e^2)*sqrt(1+c^2*x^2)))/(e*(c^2*d^2+e^2)^(3/2))-1/2*b*c*sqrt(1+c^2*x^2)/((c^2*d^2+e^2)*(d+e*x))],
[(a+b*arcsinh(c*x))/(d+e*x)^4,x,5,1/3*(-a-b*arcsinh(c*x))/(e*(d+e*x)^3)-1/6*b*c^3*(2*c^2*d^2-e^2)*arctanh((e-c^2*d*x)/(sqrt(c^2*d^2+e^2)*sqrt(1+c^2*x^2)))/(e*(c^2*d^2+e^2)^(5/2))-1/6*b*c*sqrt(1+c^2*x^2)/((c^2*d^2+e^2)*(d+e*x)^2)-1/2*b*c^3*d*sqrt(1+c^2*x^2)/((c^2*d^2+e^2)^2*(d+e*x))],
[(d+e*x)^3*(a+b*arcsinh(c*x))^2,x,18,2*b^2*d^3*x-4/3*b^2*d*e^2*x/c^2+3/4*b^2*d^2*e*x^2-3/32*b^2*e^3*x^2/c^2+2/9*b^2*d*e^2*x^3+1/32*b^2*e^3*x^4-1/4*d^4*(a+b*arcsinh(c*x))^2/e+3/4*d^2*e*(a+b*arcsinh(c*x))^2/c^2-3/32*e^3*(a+b*arcsinh(c*x))^2/c^4+1/4*(d+e*x)^4*(a+b*arcsinh(c*x))^2/e-2*b*d^3*(a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)/c+4/3*b*d*e^2*(a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)/c^3-3/2*b*d^2*e*x*(a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)/c+3/16*b*e^3*x*(a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)/c^3-2/3*b*d*e^2*x^2*(a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)/c-1/8*b*e^3*x^3*(a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)/c],
[(d+e*x)^2*(a+b*arcsinh(c*x))^2,x,13,2*b^2*d^2*x-4/9*b^2*e^2*x/c^2+1/2*b^2*d*e*x^2+2/27*b^2*e^2*x^3-1/3*d^3*(a+b*arcsinh(c*x))^2/e+1/2*d*e*(a+b*arcsinh(c*x))^2/c^2+1/3*(d+e*x)^3*(a+b*arcsinh(c*x))^2/e-2*b*d^2*(a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)/c+4/9*b*e^2*(a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)/c^3-b*d*e*x*(a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)/c-2/9*b*e^2*x^2*(a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)/c],
[(d+e*x)*(a+b*arcsinh(c*x))^2,x,9,2*b^2*d*x+1/4*b^2*e*x^2-1/2*d^2*(a+b*arcsinh(c*x))^2/e+1/4*e*(a+b*arcsinh(c*x))^2/c^2+1/2*(d+e*x)^2*(a+b*arcsinh(c*x))^2/e-2*b*d*(a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)/c-1/2*b*e*x*(a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)/c],
[(a+b*arcsinh(c*x))^2,x,3,2*b^2*x+x*(a+b*arcsinh(c*x))^2-2*b*(a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)/c],
[(a+b*arcsinh(c*x))^2/(d+e*x),x,10,-1/3*(a+b*arcsinh(c*x))^3/(b*e)+(a+b*arcsinh(c*x))^2*log(1+e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/e+(a+b*arcsinh(c*x))^2*log(1+e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/e+2*b*(a+b*arcsinh(c*x))*polylog(2,-e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/e+2*b*(a+b*arcsinh(c*x))*polylog(2,-e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/e-2*b^2*polylog(3,-e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/e-2*b^2*polylog(3,-e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/e],
[(a+b*arcsinh(c*x))^2/(d+e*x)^2,x,10,-(a+b*arcsinh(c*x))^2/(e*(d+e*x))+2*b*c*(a+b*arcsinh(c*x))*log(1+e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/(e*sqrt(c^2*d^2+e^2))-2*b*c*(a+b*arcsinh(c*x))*log(1+e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/(e*sqrt(c^2*d^2+e^2))+2*b^2*c*polylog(2,-e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/(e*sqrt(c^2*d^2+e^2))-2*b^2*c*polylog(2,-e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/(e*sqrt(c^2*d^2+e^2))],
[(a+b*arcsinh(c*x))^2/(d+e*x)^3,x,13,-1/2*(a+b*arcsinh(c*x))^2/(e*(d+e*x)^2)+b^2*c^2*log(d+e*x)/(e*(c^2*d^2+e^2))+b*c^3*d*(a+b*arcsinh(c*x))*log(1+e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/(e*(c^2*d^2+e^2)^(3/2))-b*c^3*d*(a+b*arcsinh(c*x))*log(1+e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/(e*(c^2*d^2+e^2)^(3/2))+b^2*c^3*d*polylog(2,-e*exp(arcsinh(c*x))/(c*d-sqrt(c^2*d^2+e^2)))/(e*(c^2*d^2+e^2)^(3/2))-b^2*c^3*d*polylog(2,-e*exp(arcsinh(c*x))/(c*d+sqrt(c^2*d^2+e^2)))/(e*(c^2*d^2+e^2)^(3/2))-b*c*(a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)/((c^2*d^2+e^2)*(d+e*x))],

# n<0
[(d+e*x)^3/(a+b*arcsinh(c*x)),x,27,d^3*Chi(a/b+arcsinh(c*x))*cosh(a/b)/(b*c)-3/4*d*e^2*Chi(a/b+arcsinh(c*x))*cosh(a/b)/(b*c^3)+3/4*d*e^2*Chi(3*a/b+3*arcsinh(c*x))*cosh(3*a/b)/(b*c^3)+3/2*d^2*e*cosh(2*a/b)*Shi(2*a/b+2*arcsinh(c*x))/(b*c^2)-1/4*e^3*cosh(2*a/b)*Shi(2*a/b+2*arcsinh(c*x))/(b*c^4)+1/8*e^3*cosh(4*a/b)*Shi(4*a/b+4*arcsinh(c*x))/(b*c^4)-d^3*Shi(a/b+arcsinh(c*x))*sinh(a/b)/(b*c)+3/4*d*e^2*Shi(a/b+arcsinh(c*x))*sinh(a/b)/(b*c^3)-3/2*d^2*e*Chi(2*a/b+2*arcsinh(c*x))*sinh(2*a/b)/(b*c^2)+1/4*e^3*Chi(2*a/b+2*arcsinh(c*x))*sinh(2*a/b)/(b*c^4)-3/4*d*e^2*Shi(3*a/b+3*arcsinh(c*x))*sinh(3*a/b)/(b*c^3)-1/8*e^3*Chi(4*a/b+4*arcsinh(c*x))*sinh(4*a/b)/(b*c^4)],
[(d+e*x)^2/(a+b*arcsinh(c*x)),x,17,d^2*Chi(a/b+arcsinh(c*x))*cosh(a/b)/(b*c)-1/4*e^2*Chi(a/b+arcsinh(c*x))*cosh(a/b)/(b*c^3)+1/4*e^2*Chi(3*a/b+3*arcsinh(c*x))*cosh(3*a/b)/(b*c^3)+d*e*cosh(2*a/b)*Shi(2*a/b+2*arcsinh(c*x))/(b*c^2)-d^2*Shi(a/b+arcsinh(c*x))*sinh(a/b)/(b*c)+1/4*e^2*Shi(a/b+arcsinh(c*x))*sinh(a/b)/(b*c^3)-d*e*Chi(2*a/b+2*arcsinh(c*x))*sinh(2*a/b)/(b*c^2)-1/4*e^2*Shi(3*a/b+3*arcsinh(c*x))*sinh(3*a/b)/(b*c^3)],
[(d+e*x)/(a+b*arcsinh(c*x)),x,11,d*Chi(a/b+arcsinh(c*x))*cosh(a/b)/(b*c)+1/2*e*cosh(2*a/b)*Shi(2*a/b+2*arcsinh(c*x))/(b*c^2)-d*Shi(a/b+arcsinh(c*x))*sinh(a/b)/(b*c)-1/2*e*Chi(2*a/b+2*arcsinh(c*x))*sinh(2*a/b)/(b*c^2)],
[1/(a+b*arcsinh(c*x)),x,4,Chi((a+b*arcsinh(c*x))/b)*cosh(a/b)/(b*c)-Shi((a+b*arcsinh(c*x))/b)*sinh(a/b)/(b*c)],
[1/((d+e*x)*(a+b*arcsinh(c*x))),x,0,Unintegrable(1/((d+e*x)*(a+b*arcsinh(c*x))),x)],
[1/((d+e*x)^2*(a+b*arcsinh(c*x))),x,0,Unintegrable(1/((d+e*x)^2*(a+b*arcsinh(c*x))),x)],
[(d+e*x)^2/(a+b*arcsinh(c*x))^2,x,19,2*d*e*Chi(2*(a+b*arcsinh(c*x))/b)*cosh(2*a/b)/(b^2*c^2)+d^2*cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/(b^2*c)-1/4*e^2*cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/(b^2*c^3)+3/4*e^2*cosh(3*a/b)*Shi(3*(a+b*arcsinh(c*x))/b)/(b^2*c^3)-d^2*Chi((a+b*arcsinh(c*x))/b)*sinh(a/b)/(b^2*c)+1/4*e^2*Chi((a+b*arcsinh(c*x))/b)*sinh(a/b)/(b^2*c^3)-2*d*e*Shi(2*(a+b*arcsinh(c*x))/b)*sinh(2*a/b)/(b^2*c^2)-3/4*e^2*Chi(3*(a+b*arcsinh(c*x))/b)*sinh(3*a/b)/(b^2*c^3)-d^2*sqrt(1+c^2*x^2)/(b*c*(a+b*arcsinh(c*x)))-2*d*e*x*sqrt(1+c^2*x^2)/(b*c*(a+b*arcsinh(c*x)))-e^2*x^2*sqrt(1+c^2*x^2)/(b*c*(a+b*arcsinh(c*x)))],
[(d+e*x)/(a+b*arcsinh(c*x))^2,x,11,e*Chi(2*(a+b*arcsinh(c*x))/b)*cosh(2*a/b)/(b^2*c^2)+d*cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/(b^2*c)-d*Chi((a+b*arcsinh(c*x))/b)*sinh(a/b)/(b^2*c)-e*Shi(2*(a+b*arcsinh(c*x))/b)*sinh(2*a/b)/(b^2*c^2)-d*sqrt(1+c^2*x^2)/(b*c*(a+b*arcsinh(c*x)))-e*x*sqrt(1+c^2*x^2)/(b*c*(a+b*arcsinh(c*x)))],
[1/(a+b*arcsinh(c*x))^2,x,5,cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/(b^2*c)-Chi((a+b*arcsinh(c*x))/b)*sinh(a/b)/(b^2*c)-sqrt(1+c^2*x^2)/(b*c*(a+b*arcsinh(c*x)))],
[1/((d+e*x)*(a+b*arcsinh(c*x))^2),x,0,Unintegrable(1/((d+e*x)*(a+b*arcsinh(c*x))^2),x)],
[1/((d+e*x)^2*(a+b*arcsinh(c*x))^2),x,0,Unintegrable(1/((d+e*x)^2*(a+b*arcsinh(c*x))^2),x)],

# Integrands of the form (d+e x)^m (a+b ArcSinh[c x])^(n/2)

# Integrands of the form (d+e x)^m (a+b ArcSinh[c x])^n with m symbolic
[(d+e*x)^m*(a+b*arcsinh(c*x))^2,x,1,(d+e*x)^(1+m)*(a+b*arcsinh(c*x))^2/(e*(1+m))-2*b*c*Unintegrable((d+e*x)^(1+m)*(a+b*arcsinh(c*x))/sqrt(1+c^2*x^2),x)/(e*(1+m))],
[(d+e*x)^m*(a+b*arcsinh(c*x)),x,3,(d+e*x)^(1+m)*(a+b*arcsinh(c*x))/(e*(1+m))-b*c*(d+e*x)^(2+m)*AppellF1(2+m,1/2,1/2,3+m,(d+e*x)/(d-e/sqrt(-c^2)),(d+e*x)/(d+e/sqrt(-c^2)))*sqrt(1+(-d-e*x)/(d-e/sqrt(-c^2)))*sqrt(1+(-d-e*x)/(d+e/sqrt(-c^2)))/(e^2*(1+m)*(2+m)*sqrt(1+c^2*x^2))],
[(d+e*x)^m/(a+b*arcsinh(c*x)),x,0,Unintegrable((d+e*x)^m/(a+b*arcsinh(c*x)),x)],
[(d+e*x)^m/(a+b*arcsinh(c*x))^2,x,0,Unintegrable((d+e*x)^m/(a+b*arcsinh(c*x))^2,x)],

# Integrands of the form (f+g x)^m (d+e x^2)^p (a+b ArcSinh[c x])^n where e=c^2 d

# Integrands of the form (f+g x)^m (d+c^2 d x^2)^(p/2) (a+b ArcSinh[c x])^n

# p>0
[(f+g*x)^3*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2),x,16,1/2*f^3*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+3/8*f*g^2*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2+3/4*f*g^2*x^3*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+f^2*g*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2-1/3*g^3*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^4+1/5*g^3*(1+c^2*x^2)^2*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^4-b*f^2*g*x*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))+2/15*b*g^3*x*sqrt(d+c^2*d*x^2)/(c^3*sqrt(1+c^2*x^2))-1/4*b*c*f^3*x^2*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-3/16*b*f*g^2*x^2*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-1/3*b*c*f^2*g*x^3*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/45*b*g^3*x^3*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-3/16*b*c*f*g^2*x^4*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/25*b*c*g^3*x^5*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)+1/4*f^3*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c*sqrt(1+c^2*x^2))-3/16*f*g^2*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c^3*sqrt(1+c^2*x^2))],
[(f+g*x)^2*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2),x,13,1/2*f^2*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+1/8*g^2*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2+1/4*g^2*x^3*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+2/3*f*g*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2-2/3*b*f*g*x*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-1/4*b*c*f^2*x^2*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/16*b*g^2*x^2*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-2/9*b*c*f*g*x^3*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/16*b*c*g^2*x^4*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)+1/4*f^2*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c*sqrt(1+c^2*x^2))-1/16*g^2*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c^3*sqrt(1+c^2*x^2))],
[(f+g*x)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2),x,8,1/2*f*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+1/3*g*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2-1/3*b*g*x*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-1/4*b*c*f*x^2*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/9*b*c*g*x^3*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)+1/4*f*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c*sqrt(1+c^2*x^2))],
[(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/(f+g*x),x,22,a*sqrt(d+c^2*d*x^2)/g+b*arcsinh(c*x)*sqrt(d+c^2*d*x^2)/g-b*c*x*sqrt(d+c^2*d*x^2)/(g*sqrt(1+c^2*x^2))-1/2*c*x*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*g*sqrt(1+c^2*x^2))-1/2*(1+c^2*f^2/g^2)*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c*(f+g*x)*sqrt(1+c^2*x^2))-a*arctanh((g-c^2*f*x)/(sqrt(c^2*f^2+g^2)*sqrt(1+c^2*x^2)))*sqrt(c^2*f^2+g^2)*sqrt(d+c^2*d*x^2)/(g^2*sqrt(1+c^2*x^2))+b*arcsinh(c*x)*log(1+exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))*sqrt(c^2*f^2+g^2)*sqrt(d+c^2*d*x^2)/(g^2*sqrt(1+c^2*x^2))-b*arcsinh(c*x)*log(1+exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))*sqrt(c^2*f^2+g^2)*sqrt(d+c^2*d*x^2)/(g^2*sqrt(1+c^2*x^2))+b*polylog(2,-exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))*sqrt(c^2*f^2+g^2)*sqrt(d+c^2*d*x^2)/(g^2*sqrt(1+c^2*x^2))-b*polylog(2,-exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))*sqrt(c^2*f^2+g^2)*sqrt(d+c^2*d*x^2)/(g^2*sqrt(1+c^2*x^2))+1/2*(a+b*arcsinh(c*x))^2*sqrt(1+c^2*x^2)*sqrt(d+c^2*d*x^2)/(b*c*(f+g*x))],
[(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/(f+g*x)^2,x,35,-a*sqrt(d+c^2*d*x^2)/(g*(f+g*x))-b*arcsinh(c*x)*sqrt(d+c^2*d*x^2)/(g*(f+g*x))+a*c^3*f^2*arcsinh(c*x)*sqrt(d+c^2*d*x^2)/(g^2*(c^2*f^2+g^2)*sqrt(1+c^2*x^2))+1/2*b*c^3*f^2*arcsinh(c*x)^2*sqrt(d+c^2*d*x^2)/(g^2*(c^2*f^2+g^2)*sqrt(1+c^2*x^2))-1/2*(g-c^2*f*x)^2*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c*(c^2*f^2+g^2)*(f+g*x)^2*sqrt(1+c^2*x^2))+b*c*log(f+g*x)*sqrt(d+c^2*d*x^2)/(g^2*sqrt(1+c^2*x^2))+a*c^2*f*arctanh((g-c^2*f*x)/(sqrt(c^2*f^2+g^2)*sqrt(1+c^2*x^2)))*sqrt(d+c^2*d*x^2)/(g^2*sqrt(c^2*f^2+g^2)*sqrt(1+c^2*x^2))-b*c^2*f*arcsinh(c*x)*log(1+exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))*sqrt(d+c^2*d*x^2)/(g^2*sqrt(c^2*f^2+g^2)*sqrt(1+c^2*x^2))+b*c^2*f*arcsinh(c*x)*log(1+exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))*sqrt(d+c^2*d*x^2)/(g^2*sqrt(c^2*f^2+g^2)*sqrt(1+c^2*x^2))-b*c^2*f*polylog(2,-exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))*sqrt(d+c^2*d*x^2)/(g^2*sqrt(c^2*f^2+g^2)*sqrt(1+c^2*x^2))+b*c^2*f*polylog(2,-exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))*sqrt(d+c^2*d*x^2)/(g^2*sqrt(c^2*f^2+g^2)*sqrt(1+c^2*x^2))+1/2*(a+b*arcsinh(c*x))^2*sqrt(1+c^2*x^2)*sqrt(d+c^2*d*x^2)/(b*c*(f+g*x)^2)],
[(f+g*x)^3*(d+c^2*d*x^2)^(3/2)*(a+b*arcsinh(c*x)),x,24,3/8*d*f^3*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+3/16*d*f*g^2*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2+3/8*d*f*g^2*x^3*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+1/4*d*f^3*x*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+1/2*d*f*g^2*x^3*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+3/5*d*f^2*g*(1+c^2*x^2)^2*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2-1/5*d*g^3*(1+c^2*x^2)^2*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^4+1/7*d*g^3*(1+c^2*x^2)^3*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^4-3/5*b*d*f^2*g*x*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))+2/35*b*d*g^3*x*sqrt(d+c^2*d*x^2)/(c^3*sqrt(1+c^2*x^2))-5/16*b*c*d*f^3*x^2*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-3/32*b*d*f*g^2*x^2*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-2/5*b*c*d*f^2*g*x^3*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/105*b*d*g^3*x^3*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-1/16*b*c^3*d*f^3*x^4*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-7/32*b*c*d*f*g^2*x^4*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-3/25*b*c^3*d*f^2*g*x^5*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-8/175*b*c*d*g^3*x^5*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/12*b*c^3*d*f*g^2*x^6*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/49*b*c^3*d*g^3*x^7*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)+3/16*d*f^3*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c*sqrt(1+c^2*x^2))-3/32*d*f*g^2*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c^3*sqrt(1+c^2*x^2))],
[(f+g*x)^2*(d+c^2*d*x^2)^(3/2)*(a+b*arcsinh(c*x)),x,20,3/8*d*f^2*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+1/16*d*g^2*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2+1/8*d*g^2*x^3*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+1/4*d*f^2*x*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+1/6*d*g^2*x^3*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+2/5*d*f*g*(1+c^2*x^2)^2*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2-2/5*b*d*f*g*x*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-5/16*b*c*d*f^2*x^2*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/32*b*d*g^2*x^2*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-4/15*b*c*d*f*g*x^3*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/16*b*c^3*d*f^2*x^4*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-7/96*b*c*d*g^2*x^4*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-2/25*b*c^3*d*f*g*x^5*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/36*b*c^3*d*g^2*x^6*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)+3/16*d*f^2*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c*sqrt(1+c^2*x^2))-1/32*d*g^2*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c^3*sqrt(1+c^2*x^2))],
[(f+g*x)*(d+c^2*d*x^2)^(3/2)*(a+b*arcsinh(c*x)),x,12,3/8*d*f*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+1/4*d*f*x*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+1/5*d*g*(1+c^2*x^2)^2*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2-1/5*b*d*g*x*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-5/16*b*c*d*f*x^2*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-2/15*b*c*d*g*x^3*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/16*b*c^3*d*f*x^4*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/25*b*c^3*d*g*x^5*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)+3/16*d*f*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c*sqrt(1+c^2*x^2))],
[(d+c^2*d*x^2)^(3/2)*(a+b*arcsinh(c*x))/(f+g*x),x,29,a*d*(c^2*f^2+g^2)*sqrt(d+c^2*d*x^2)/g^3+b*d*(c^2*f^2+g^2)*arcsinh(c*x)*sqrt(d+c^2*d*x^2)/g^3-1/2*c^2*d*f*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/g^2+1/3*d*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/g-1/3*b*c*d*x*sqrt(d+c^2*d*x^2)/(g*sqrt(1+c^2*x^2))-b*c*d*(c^2*f^2+g^2)*x*sqrt(d+c^2*d*x^2)/(g^3*sqrt(1+c^2*x^2))+1/4*b*c^3*d*f*x^2*sqrt(d+c^2*d*x^2)/(g^2*sqrt(1+c^2*x^2))-1/9*b*c^3*d*x^3*sqrt(d+c^2*d*x^2)/(g*sqrt(1+c^2*x^2))-1/4*c*d*f*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*g^2*sqrt(1+c^2*x^2))-1/2*c*d*(c^2*f^2+g^2)*x*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*g^3*sqrt(1+c^2*x^2))-1/2*d*(c^2*f^2+g^2)^2*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c*g^4*(f+g*x)*sqrt(1+c^2*x^2))-a*d*(c^2*f^2+g^2)^(3/2)*arctanh((g-c^2*f*x)/(sqrt(c^2*f^2+g^2)*sqrt(1+c^2*x^2)))*sqrt(d+c^2*d*x^2)/(g^4*sqrt(1+c^2*x^2))+b*d*(c^2*f^2+g^2)^(3/2)*arcsinh(c*x)*log(1+exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))*sqrt(d+c^2*d*x^2)/(g^4*sqrt(1+c^2*x^2))-b*d*(c^2*f^2+g^2)^(3/2)*arcsinh(c*x)*log(1+exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))*sqrt(d+c^2*d*x^2)/(g^4*sqrt(1+c^2*x^2))+b*d*(c^2*f^2+g^2)^(3/2)*polylog(2,-exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))*sqrt(d+c^2*d*x^2)/(g^4*sqrt(1+c^2*x^2))-b*d*(c^2*f^2+g^2)^(3/2)*polylog(2,-exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))*sqrt(d+c^2*d*x^2)/(g^4*sqrt(1+c^2*x^2))+1/2*d*(c^2*f^2+g^2)*(a+b*arcsinh(c*x))^2*sqrt(1+c^2*x^2)*sqrt(d+c^2*d*x^2)/(b*c*g^2*(f+g*x))],

#  {(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])/(f + g*x)^2, x, 71, -((2*a*c^2*d*f*Sqrt[d + c^2*d*x^2])/g^3) - (a*d*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2])/(g^3*(f + g*x)) + (2*b*c^3*d*f*x*Sqrt[d + c^2*d*x^2])/(g^3*Sqrt[1 + c^2*x^2]) - (b*c^3*d*x^2*Sqrt[d + c^2*d*x^2])/(4*g^2*Sqrt[1 + c^2*x^2]) - (2*b*c^2*d*f*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/g^3 - (b*d*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(g^3*(f + g*x)) + (a*c^3*d*f^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(g^4*Sqrt[1 + c^2*x^2]) + (b*c^3*d*f^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2)/(2*g^4*Sqrt[1 + c^2*x^2]) + (c^2*d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*g^2) + (c*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*g^2*Sqrt[1 + c^2*x^2]) + (c^3*d*f*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(b*g^3*Sqrt[1 + c^2*x^2]) - (d*(g - c^2*f*x)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^2*(f + g*x)^2*Sqrt[1 + c^2*x^2]) + (c*d*f*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(b*g^4*(f + g*x)*Sqrt[1 + c^2*x^2]) + (d*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^2*(f + g*x)^2) - (c*d*f*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(b*g^2*(f + g*x)) + (3*a*c^2*d*f*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*ArcTanh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])])/(g^4*Sqrt[1 + c^2*x^2]) - (3*b*c^2*d*f*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(g^4*Sqrt[1 + c^2*x^2]) + (3*b*c^2*d*f*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(g^4*Sqrt[1 + c^2*x^2]) + (b*c*d*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2]*Log[f + g*x])/(g^4*Sqrt[1 + c^2*x^2]) - (3*b*c^2*d*f*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/(g^4*Sqrt[1 + c^2*x^2]) + (3*b*c^2*d*f*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/(g^4*Sqrt[1 + c^2*x^2])} 
[(f+g*x)^3*(d+c^2*d*x^2)^(5/2)*(a+b*arcsinh(c*x)),x,30,-1/36*b*d^2*f^3*(1+c^2*x^2)^(5/2)*sqrt(d+c^2*d*x^2)/c+5/16*d^2*f^3*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+15/128*d^2*f*g^2*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2+15/64*d^2*f*g^2*x^3*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+5/24*d^2*f^3*x*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+5/16*d^2*f*g^2*x^3*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+1/6*d^2*f^3*x*(1+c^2*x^2)^2*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+3/8*d^2*f*g^2*x^3*(1+c^2*x^2)^2*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+3/7*d^2*f^2*g*(1+c^2*x^2)^3*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2-1/7*d^2*g^3*(1+c^2*x^2)^3*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^4+1/9*d^2*g^3*(1+c^2*x^2)^4*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^4-3/7*b*d^2*f^2*g*x*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))+2/63*b*d^2*g^3*x*sqrt(d+c^2*d*x^2)/(c^3*sqrt(1+c^2*x^2))-25/96*b*c*d^2*f^3*x^2*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-15/256*b*d^2*f*g^2*x^2*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-3/7*b*c*d^2*f^2*g*x^3*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/189*b*d^2*g^3*x^3*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-5/96*b*c^3*d^2*f^3*x^4*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-59/256*b*c*d^2*f*g^2*x^4*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-9/35*b*c^3*d^2*f^2*g*x^5*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/21*b*c*d^2*g^3*x^5*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-17/96*b*c^3*d^2*f*g^2*x^6*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-3/49*b*c^5*d^2*f^2*g*x^7*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-19/441*b*c^3*d^2*g^3*x^7*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-3/64*b*c^5*d^2*f*g^2*x^8*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/81*b*c^5*d^2*g^3*x^9*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)+5/32*d^2*f^3*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c*sqrt(1+c^2*x^2))-15/256*d^2*f*g^2*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c^3*sqrt(1+c^2*x^2))],
[(f+g*x)^2*(d+c^2*d*x^2)^(5/2)*(a+b*arcsinh(c*x)),x,26,-1/36*b*d^2*f^2*(1+c^2*x^2)^(5/2)*sqrt(d+c^2*d*x^2)/c+5/16*d^2*f^2*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+5/128*d^2*g^2*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2+5/64*d^2*g^2*x^3*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+5/24*d^2*f^2*x*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+5/48*d^2*g^2*x^3*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+1/6*d^2*f^2*x*(1+c^2*x^2)^2*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+1/8*d^2*g^2*x^3*(1+c^2*x^2)^2*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+2/7*d^2*f*g*(1+c^2*x^2)^3*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2-2/7*b*d^2*f*g*x*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-25/96*b*c*d^2*f^2*x^2*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-5/256*b*d^2*g^2*x^2*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-2/7*b*c*d^2*f*g*x^3*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-5/96*b*c^3*d^2*f^2*x^4*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-59/768*b*c*d^2*g^2*x^4*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-6/35*b*c^3*d^2*f*g*x^5*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-17/288*b*c^3*d^2*g^2*x^6*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-2/49*b*c^5*d^2*f*g*x^7*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/64*b*c^5*d^2*g^2*x^8*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)+5/32*d^2*f^2*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c*sqrt(1+c^2*x^2))-5/256*d^2*g^2*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c^3*sqrt(1+c^2*x^2))],
[(f+g*x)*(d+c^2*d*x^2)^(5/2)*(a+b*arcsinh(c*x)),x,14,-1/36*b*d^2*f*(1+c^2*x^2)^(5/2)*sqrt(d+c^2*d*x^2)/c+5/16*d^2*f*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+5/24*d^2*f*x*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+1/6*d^2*f*x*(1+c^2*x^2)^2*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)+1/7*d^2*g*(1+c^2*x^2)^3*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/c^2-1/7*b*d^2*g*x*sqrt(d+c^2*d*x^2)/(c*sqrt(1+c^2*x^2))-25/96*b*c*d^2*f*x^2*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/7*b*c*d^2*g*x^3*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-5/96*b*c^3*d^2*f*x^4*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-3/35*b*c^3*d^2*g*x^5*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)-1/49*b*c^5*d^2*g*x^7*sqrt(d+c^2*d*x^2)/sqrt(1+c^2*x^2)+5/32*d^2*f*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c*sqrt(1+c^2*x^2))],
[(d+c^2*d*x^2)^(5/2)*(a+b*arcsinh(c*x))/(f+g*x),x,37,a*d^2*(c^2*f^2+g^2)^2*sqrt(d+c^2*d*x^2)/g^5+b*d^2*(c^2*f^2+g^2)^2*arcsinh(c*x)*sqrt(d+c^2*d*x^2)/g^5-1/8*c^2*d^2*f*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/g^2-1/2*c^2*d^2*f*(c^2*f^2+2*g^2)*x*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/g^4-1/4*c^4*d^2*f*x^3*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/g^2-1/3*d^2*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/g+1/3*d^2*(c^2*f^2+2*g^2)*(1+c^2*x^2)*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/g^3+1/5*d^2*(1+c^2*x^2)^2*(a+b*arcsinh(c*x))*sqrt(d+c^2*d*x^2)/g+2/15*b*c*d^2*x*sqrt(d+c^2*d*x^2)/(g*sqrt(1+c^2*x^2))-b*c*d^2*(c^2*f^2+g^2)^2*x*sqrt(d+c^2*d*x^2)/(g^5*sqrt(1+c^2*x^2))-1/3*b*c*d^2*(c^2*f^2+2*g^2)*x*sqrt(d+c^2*d*x^2)/(g^3*sqrt(1+c^2*x^2))+1/16*b*c^3*d^2*f*x^2*sqrt(d+c^2*d*x^2)/(g^2*sqrt(1+c^2*x^2))+1/4*b*c^3*d^2*f*(c^2*f^2+2*g^2)*x^2*sqrt(d+c^2*d*x^2)/(g^4*sqrt(1+c^2*x^2))-1/45*b*c^3*d^2*x^3*sqrt(d+c^2*d*x^2)/(g*sqrt(1+c^2*x^2))-1/9*b*c^3*d^2*(c^2*f^2+2*g^2)*x^3*sqrt(d+c^2*d*x^2)/(g^3*sqrt(1+c^2*x^2))+1/16*b*c^5*d^2*f*x^4*sqrt(d+c^2*d*x^2)/(g^2*sqrt(1+c^2*x^2))-1/25*b*c^5*d^2*x^5*sqrt(d+c^2*d*x^2)/(g*sqrt(1+c^2*x^2))+1/16*c*d^2*f*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*g^2*sqrt(1+c^2*x^2))-1/4*c*d^2*f*(c^2*f^2+2*g^2)*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*g^4*sqrt(1+c^2*x^2))-1/2*c*d^2*(c^2*f^2+g^2)^2*x*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*g^5*sqrt(1+c^2*x^2))-1/2*d^2*(c^2*f^2+g^2)^3*(a+b*arcsinh(c*x))^2*sqrt(d+c^2*d*x^2)/(b*c*g^6*(f+g*x)*sqrt(1+c^2*x^2))-a*d^2*(c^2*f^2+g^2)^(5/2)*arctanh((g-c^2*f*x)/(sqrt(c^2*f^2+g^2)*sqrt(1+c^2*x^2)))*sqrt(d+c^2*d*x^2)/(g^6*sqrt(1+c^2*x^2))+b*d^2*(c^2*f^2+g^2)^(5/2)*arcsinh(c*x)*log(1+exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))*sqrt(d+c^2*d*x^2)/(g^6*sqrt(1+c^2*x^2))-b*d^2*(c^2*f^2+g^2)^(5/2)*arcsinh(c*x)*log(1+exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))*sqrt(d+c^2*d*x^2)/(g^6*sqrt(1+c^2*x^2))+b*d^2*(c^2*f^2+g^2)^(5/2)*polylog(2,-exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))*sqrt(d+c^2*d*x^2)/(g^6*sqrt(1+c^2*x^2))-b*d^2*(c^2*f^2+g^2)^(5/2)*polylog(2,-exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))*sqrt(d+c^2*d*x^2)/(g^6*sqrt(1+c^2*x^2))+1/2*d^2*(c^2*f^2+g^2)^2*(a+b*arcsinh(c*x))^2*sqrt(1+c^2*x^2)*sqrt(d+c^2*d*x^2)/(b*c*g^4*(f+g*x))],

#  {(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])/(f + g*x)^2, x, 78, -((4*a*c^2*d^2*f*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2])/g^5) - (a*d^2*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2])/(g^5*(f + g*x)) + (2*b*c^3*d^2*f*x*Sqrt[d + c^2*d*x^2])/(3*g^3*Sqrt[1 + c^2*x^2]) + (4*b*c^3*d^2*f*(c^2*f^2 + g^2)*x*Sqrt[d + c^2*d*x^2])/(g^5*Sqrt[1 + c^2*x^2]) - (b*c^3*d^2*x^2*Sqrt[d + c^2*d*x^2])/(16*g^2*Sqrt[1 + c^2*x^2]) - (b*c^3*d^2*(3*c^2*f^2 + 2*g^2)*x^2*Sqrt[d + c^2*d*x^2])/(4*g^4*Sqrt[1 + c^2*x^2]) + (2*b*c^5*d^2*f*x^3*Sqrt[d + c^2*d*x^2])/(9*g^3*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*x^4*Sqrt[d + c^2*d*x^2])/(16*g^2*Sqrt[1 + c^2*x^2]) - (4*b*c^2*d^2*f*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/g^5 - (b*d^2*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(g^5*(f + g*x)) + (a*c^3*d^2*f^2*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(g^6*Sqrt[1 + c^2*x^2]) + (b*c^3*d^2*f^2*(c^2*f^2 + g^2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2)/(2*g^6*Sqrt[1 + c^2*x^2]) + (c^2*d^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*g^2) + (c^2*d^2*(3*c^2*f^2 + 2*g^2)*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(2*g^4) + (c^4*d^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(4*g^2) - (2*c^2*d^2*f*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*g^3) - (c*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*g^2*Sqrt[1 + c^2*x^2]) + (c*d^2*(3*c^2*f^2 + 2*g^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*g^4*Sqrt[1 + c^2*x^2]) + (2*c^3*d^2*f*(c^2*f^2 + g^2)*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(b*g^5*Sqrt[1 + c^2*x^2]) - (d^2*(c^2*f^2 + g^2)*(g - c^2*f*x)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^4*(f + g*x)^2*Sqrt[1 + c^2*x^2]) + (2*c*d^2*f*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(b*g^6*(f + g*x)*Sqrt[1 + c^2*x^2]) + (d^2*(c^2*f^2 + g^2)^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*g^4*(f + g*x)^2) - (2*c*d^2*f*(c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(b*g^4*(f + g*x)) + (5*a*c^2*d^2*f*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*ArcTanh[(g - c^2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])])/(g^6*Sqrt[1 + c^2*x^2]) - (5*b*c^2*d^2*f*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(g^6*Sqrt[1 + c^2*x^2]) + (5*b*c^2*d^2*f*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(g^6*Sqrt[1 + c^2*x^2]) + (b*c*d^2*(c^2*f^2 + g^2)^2*Sqrt[d + c^2*d*x^2]*Log[f + g*x])/(g^6*Sqrt[1 + c^2*x^2]) - (5*b*c^2*d^2*f*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/(g^6*Sqrt[1 + c^2*x^2]) + (5*b*c^2*d^2*f*(c^2*f^2 + g^2)^(3/2)*Sqrt[d + c^2*d*x^2]*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/(g^6*Sqrt[1 + c^2*x^2])} 

# p<0
[(f+g*x)^3*(a+b*arcsinh(c*x))/sqrt(d+c^2*d*x^2),x,13,3*f^2*g*(1+c^2*x^2)*(a+b*arcsinh(c*x))/(c^2*sqrt(d+c^2*d*x^2))-2/3*g^3*(1+c^2*x^2)*(a+b*arcsinh(c*x))/(c^4*sqrt(d+c^2*d*x^2))+3/2*f*g^2*x*(1+c^2*x^2)*(a+b*arcsinh(c*x))/(c^2*sqrt(d+c^2*d*x^2))+1/3*g^3*x^2*(1+c^2*x^2)*(a+b*arcsinh(c*x))/(c^2*sqrt(d+c^2*d*x^2))-3*b*f^2*g*x*sqrt(1+c^2*x^2)/(c*sqrt(d+c^2*d*x^2))+2/3*b*g^3*x*sqrt(1+c^2*x^2)/(c^3*sqrt(d+c^2*d*x^2))-3/4*b*f*g^2*x^2*sqrt(1+c^2*x^2)/(c*sqrt(d+c^2*d*x^2))-1/9*b*g^3*x^3*sqrt(1+c^2*x^2)/(c*sqrt(d+c^2*d*x^2))+1/2*f^3*(a+b*arcsinh(c*x))^2*sqrt(1+c^2*x^2)/(b*c*sqrt(d+c^2*d*x^2))-3/4*f*g^2*(a+b*arcsinh(c*x))^2*sqrt(1+c^2*x^2)/(b*c^3*sqrt(d+c^2*d*x^2))],
[(f+g*x)^2*(a+b*arcsinh(c*x))/sqrt(d+c^2*d*x^2),x,9,2*f*g*(1+c^2*x^2)*(a+b*arcsinh(c*x))/(c^2*sqrt(d+c^2*d*x^2))+1/2*g^2*x*(1+c^2*x^2)*(a+b*arcsinh(c*x))/(c^2*sqrt(d+c^2*d*x^2))-2*b*f*g*x*sqrt(1+c^2*x^2)/(c*sqrt(d+c^2*d*x^2))-1/4*b*g^2*x^2*sqrt(1+c^2*x^2)/(c*sqrt(d+c^2*d*x^2))+1/2*f^2*(a+b*arcsinh(c*x))^2*sqrt(1+c^2*x^2)/(b*c*sqrt(d+c^2*d*x^2))-1/4*g^2*(a+b*arcsinh(c*x))^2*sqrt(1+c^2*x^2)/(b*c^3*sqrt(d+c^2*d*x^2))],
[(f+g*x)*(a+b*arcsinh(c*x))/sqrt(d+c^2*d*x^2),x,6,g*(1+c^2*x^2)*(a+b*arcsinh(c*x))/(c^2*sqrt(d+c^2*d*x^2))-b*g*x*sqrt(1+c^2*x^2)/(c*sqrt(d+c^2*d*x^2))+1/2*f*(a+b*arcsinh(c*x))^2*sqrt(1+c^2*x^2)/(b*c*sqrt(d+c^2*d*x^2))],
[(a+b*arcsinh(c*x))/sqrt(d+c^2*d*x^2),x,1,1/2*(a+b*arcsinh(c*x))^2*sqrt(1+c^2*x^2)/(b*c*sqrt(d+c^2*d*x^2))],
[(a+b*arcsinh(c*x))/((f+g*x)*sqrt(d+c^2*d*x^2)),x,10,(a+b*arcsinh(c*x))*log(1+exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))*sqrt(1+c^2*x^2)/(sqrt(c^2*f^2+g^2)*sqrt(d+c^2*d*x^2))-(a+b*arcsinh(c*x))*log(1+exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))*sqrt(1+c^2*x^2)/(sqrt(c^2*f^2+g^2)*sqrt(d+c^2*d*x^2))+b*polylog(2,-exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))*sqrt(1+c^2*x^2)/(sqrt(c^2*f^2+g^2)*sqrt(d+c^2*d*x^2))-b*polylog(2,-exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))*sqrt(1+c^2*x^2)/(sqrt(c^2*f^2+g^2)*sqrt(d+c^2*d*x^2))],
[(a+b*arcsinh(c*x))/((f+g*x)^2*sqrt(d+c^2*d*x^2)),x,13,-g*(1+c^2*x^2)*(a+b*arcsinh(c*x))/((c^2*f^2+g^2)*(f+g*x)*sqrt(d+c^2*d*x^2))+b*c*log(f+g*x)*sqrt(1+c^2*x^2)/((c^2*f^2+g^2)*sqrt(d+c^2*d*x^2))+c^2*f*(a+b*arcsinh(c*x))*log(1+exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))*sqrt(1+c^2*x^2)/((c^2*f^2+g^2)^(3/2)*sqrt(d+c^2*d*x^2))-c^2*f*(a+b*arcsinh(c*x))*log(1+exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))*sqrt(1+c^2*x^2)/((c^2*f^2+g^2)^(3/2)*sqrt(d+c^2*d*x^2))+b*c^2*f*polylog(2,-exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))*sqrt(1+c^2*x^2)/((c^2*f^2+g^2)^(3/2)*sqrt(d+c^2*d*x^2))-b*c^2*f*polylog(2,-exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))*sqrt(1+c^2*x^2)/((c^2*f^2+g^2)^(3/2)*sqrt(d+c^2*d*x^2))],

# Integrands of the form Log[h (f + g x)^m] (d+e x^2)^p (a+b ArcSinh[c x])^n

# Integrands of the form Log[h (f + g x)^m] (d+c^2 d x^2)^(p/2) (a+b ArcSinh[c x])^n

# p>0

# p<0
[(a+b*arcsinh(c*x))^n*log(h*(f+g*x)^m)/sqrt(1+c^2*x^2),x,0,Unintegrable((a+b*arcsinh(c*x))^n*log(h*(f+g*x)^m)/sqrt(1+c^2*x^2),x)],

#  {Log[h*(f + g*x)^m]*(a + b*ArcSinh[c*x])^3/Sqrt[1 + c^2*x^2], x, 18, (m*(a + b*ArcSinh[c*x])^5)/(20*b^2*c) - (m*(a + b*ArcSinh[c*x])^4*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])])/(4*b*c) - (m*(a + b*ArcSinh[c*x])^4*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/(4*b*c) + ((a + b*ArcSinh[c*x])^4*Log[h*(f + g*x)^m])/(4*b*c) - (m*(a + b*ArcSinh[c*x])^3*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c - (m*(a + b*ArcSinh[c*x])^3*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c + (3*b*m*(a + b*ArcSinh[c*x])^2*PolyLog[3, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c + (3*b*m*(a + b*ArcSinh[c*x])^2*PolyLog[3, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c - (6*b^2*m*(a + b*ArcSinh[c*x])*PolyLog[4, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c - (6*b^2*m*(a + b*ArcSinh[c*x])*PolyLog[4, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c + (6*b^3*m*PolyLog[5, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c + (6*b^3*m*PolyLog[5, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c} 
[(a+b*arcsinh(c*x))^2*log(h*(f+g*x)^m)/sqrt(1+c^2*x^2),x,13,1/12*m*(a+b*arcsinh(c*x))^4/(b^2*c)+1/3*(a+b*arcsinh(c*x))^3*log(h*(f+g*x)^m)/(b*c)-1/3*m*(a+b*arcsinh(c*x))^3*log(1+exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))/(b*c)-1/3*m*(a+b*arcsinh(c*x))^3*log(1+exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))/(b*c)-m*(a+b*arcsinh(c*x))^2*polylog(2,-exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))/c-m*(a+b*arcsinh(c*x))^2*polylog(2,-exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))/c+2*b*m*(a+b*arcsinh(c*x))*polylog(3,-exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))/c+2*b*m*(a+b*arcsinh(c*x))*polylog(3,-exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))/c-2*b^2*m*polylog(4,-exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))/c-2*b^2*m*polylog(4,-exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))/c],
[(a+b*arcsinh(c*x))*log(h*(f+g*x)^m)/sqrt(1+c^2*x^2),x,11,1/6*m*(a+b*arcsinh(c*x))^3/(b^2*c)+1/2*(a+b*arcsinh(c*x))^2*log(h*(f+g*x)^m)/(b*c)-1/2*m*(a+b*arcsinh(c*x))^2*log(1+exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))/(b*c)-1/2*m*(a+b*arcsinh(c*x))^2*log(1+exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))/(b*c)-m*(a+b*arcsinh(c*x))*polylog(2,-exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))/c-m*(a+b*arcsinh(c*x))*polylog(2,-exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))/c+b*m*polylog(3,-exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))/c+b*m*polylog(3,-exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))/c],
[log(h*(f+g*x)^m)/sqrt(1+c^2*x^2),x,9,1/2*m*arcsinh(c*x)^2/c+arcsinh(c*x)*log(h*(f+g*x)^m)/c-m*arcsinh(c*x)*log(1+exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))/c-m*arcsinh(c*x)*log(1+exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))/c-m*polylog(2,-exp(arcsinh(c*x))*g/(c*f-sqrt(c^2*f^2+g^2)))/c-m*polylog(2,-exp(arcsinh(c*x))*g/(c*f+sqrt(c^2*f^2+g^2)))/c],
[log(h*(f+g*x)^m)/((a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)),x,0,Unintegrable(log(h*(f+g*x)^m)/((a+b*arcsinh(c*x))*sqrt(1+c^2*x^2)),x)],

# Integrands Involving Inverse Hyperbolic Sines

# Integrands of the form u (a+b ArcSinh[c +d x])^n

# Integrands of the form x^m (a+b ArcSinh[c+d x]^n

# n>0
[x^3*arcsinh(a+b*x),x,6,-1/32*(3-24*a^2+8*a^4)*arcsinh(a+b*x)/b^4+1/4*x^4*arcsinh(a+b*x)+7/48*a*x^2*sqrt(1+(a+b*x)^2)/b^2-1/16*x^3*sqrt(1+(a+b*x)^2)/b-1/96*(4*a*(16-19*a^2)-(9-26*a^2)*(a+b*x))*sqrt(1+(a+b*x)^2)/b^4],
[x^2*arcsinh(a+b*x),x,5,-1/6*a*(3-2*a^2)*arcsinh(a+b*x)/b^3+1/3*x^3*arcsinh(a+b*x)-1/9*x^2*sqrt(1+(a+b*x)^2)/b+1/18*(4-11*a^2+5*a*b*x)*sqrt(1+(a+b*x)^2)/b^3],
[x*arcsinh(a+b*x),x,5,1/4*(1-2*a^2)*arcsinh(a+b*x)/b^2+1/2*x^2*arcsinh(a+b*x)+3/4*a*sqrt(1+(a+b*x)^2)/b^2-1/4*x*sqrt(1+(a+b*x)^2)/b],
[arcsinh(a+b*x),x,3,(a+b*x)*arcsinh(a+b*x)/b-sqrt(1+(a+b*x)^2)/b],
[arcsinh(a+b*x)/x,x,9,-1/2*arcsinh(a+b*x)^2+arcsinh(a+b*x)*log(1-exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))+arcsinh(a+b*x)*log(1-exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))+polylog(2,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))+polylog(2,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))],
[arcsinh(a+b*x)/x^2,x,4,-arcsinh(a+b*x)/x-b*arctanh((1+a*(a+b*x))/(sqrt(1+a^2)*sqrt(1+(a+b*x)^2)))/sqrt(1+a^2)],
[arcsinh(a+b*x)/x^3,x,5,-1/2*arcsinh(a+b*x)/x^2+1/2*a*b^2*arctanh((1+a*(a+b*x))/(sqrt(1+a^2)*sqrt(1+(a+b*x)^2)))/(1+a^2)^(3/2)-1/2*b*sqrt(1+(a+b*x)^2)/((1+a^2)*x)],
[arcsinh(a+b*x)/x^4,x,6,-1/3*arcsinh(a+b*x)/x^3+1/6*(1-2*a^2)*b^3*arctanh((1+a*(a+b*x))/(sqrt(1+a^2)*sqrt(1+(a+b*x)^2)))/(1+a^2)^(5/2)-1/6*b*sqrt(1+(a+b*x)^2)/((1+a^2)*x^2)+1/2*a*b^2*sqrt(1+(a+b*x)^2)/((1+a^2)^2*x)],
[arcsinh(a+b*x)/x^5,x,7,-1/4*arcsinh(a+b*x)/x^4-1/8*a*(3-2*a^2)*b^4*arctanh((1+a*(a+b*x))/(sqrt(1+a^2)*sqrt(1+(a+b*x)^2)))/(1+a^2)^(7/2)-1/12*b*sqrt(1+(a+b*x)^2)/((1+a^2)*x^3)+5/24*a*b^2*sqrt(1+(a+b*x)^2)/((1+a^2)^2*x^2)+1/24*(4-11*a^2)*b^3*sqrt(1+(a+b*x)^2)/((1+a^2)^3*x)],
[x^3*arcsinh(a+b*x)^2,x,19,4/3*a*x/b^3-2*a^3*x/b^3-3/32*(a+b*x)^2/b^4+3/4*a^2*(a+b*x)^2/b^4-2/9*a*(a+b*x)^3/b^4+1/32*(a+b*x)^4/b^4-3/32*arcsinh(a+b*x)^2/b^4+3/4*a^2*arcsinh(a+b*x)^2/b^4-1/4*a^4*arcsinh(a+b*x)^2/b^4+1/4*x^4*arcsinh(a+b*x)^2-4/3*a*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b^4+2*a^3*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b^4+3/16*(a+b*x)*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b^4-3/2*a^2*(a+b*x)*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b^4+2/3*a*(a+b*x)^2*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b^4-1/8*(a+b*x)^3*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b^4],
[x^2*arcsinh(a+b*x)^2,x,14,-4/9*x/b^2+2*a^2*x/b^2-1/2*a*(a+b*x)^2/b^3+2/27*(a+b*x)^3/b^3-1/2*a*arcsinh(a+b*x)^2/b^3+1/3*a^3*arcsinh(a+b*x)^2/b^3+1/3*x^3*arcsinh(a+b*x)^2+4/9*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b^3-2*a^2*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b^3+a*(a+b*x)*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b^3-2/9*(a+b*x)^2*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b^3],
[x*arcsinh(a+b*x)^2,x,10,-2*a*x/b+1/4*(a+b*x)^2/b^2+1/4*arcsinh(a+b*x)^2/b^2-1/2*a^2*arcsinh(a+b*x)^2/b^2+1/2*x^2*arcsinh(a+b*x)^2+2*a*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b^2-1/2*(a+b*x)*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b^2],
[arcsinh(a+b*x)^2,x,4,2*x+(a+b*x)*arcsinh(a+b*x)^2/b-2*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b],
[arcsinh(a+b*x)^2/x,x,11,-1/3*arcsinh(a+b*x)^3+arcsinh(a+b*x)^2*log(1-exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))+arcsinh(a+b*x)^2*log(1-exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))+2*arcsinh(a+b*x)*polylog(2,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))+2*arcsinh(a+b*x)*polylog(2,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))-2*polylog(3,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))-2*polylog(3,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))],
[arcsinh(a+b*x)^2/x^2,x,11,-arcsinh(a+b*x)^2/x-2*b*arcsinh(a+b*x)*log(1-exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/sqrt(1+a^2)+2*b*arcsinh(a+b*x)*log(1-exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/sqrt(1+a^2)-2*b*polylog(2,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/sqrt(1+a^2)+2*b*polylog(2,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/sqrt(1+a^2)],
[arcsinh(a+b*x)^2/x^3,x,14,-1/2*arcsinh(a+b*x)^2/x^2+b^2*log(x)/(1+a^2)+a*b^2*arcsinh(a+b*x)*log(1-exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/(1+a^2)^(3/2)-a*b^2*arcsinh(a+b*x)*log(1-exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/(1+a^2)^(3/2)+a*b^2*polylog(2,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/(1+a^2)^(3/2)-a*b^2*polylog(2,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/(1+a^2)^(3/2)-b*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/((1+a^2)*x)],
[arcsinh(a+b*x)^2/x^4,x,40,-1/3*b^2/((1+a^2)*x)-1/3*arcsinh(a+b*x)^2/x^3-a*b^3*log(x)/(1+a^2)^2-a^2*b^3*arcsinh(a+b*x)*log(1-exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/(1+a^2)^(5/2)+1/3*b^3*arcsinh(a+b*x)*log(1-exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/(1+a^2)^(3/2)+a^2*b^3*arcsinh(a+b*x)*log(1-exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/(1+a^2)^(5/2)-1/3*b^3*arcsinh(a+b*x)*log(1-exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/(1+a^2)^(3/2)-a^2*b^3*polylog(2,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/(1+a^2)^(5/2)+1/3*b^3*polylog(2,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/(1+a^2)^(3/2)+a^2*b^3*polylog(2,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/(1+a^2)^(5/2)-1/3*b^3*polylog(2,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/(1+a^2)^(3/2)-1/3*b*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/((1+a^2)*x^2)+a*b^2*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/((1+a^2)^2*x)],
[x^2*arcsinh(a+b*x)^3,x,18,-2/27*(1+(a+b*x)^2)^(3/2)/b^3-3/4*a*arcsinh(a+b*x)/b^3-4/3*(a+b*x)*arcsinh(a+b*x)/b^3+6*a^2*(a+b*x)*arcsinh(a+b*x)/b^3-3/2*a*(a+b*x)^2*arcsinh(a+b*x)/b^3+2/9*(a+b*x)^3*arcsinh(a+b*x)/b^3-1/2*a*arcsinh(a+b*x)^3/b^3+1/3*a^3*arcsinh(a+b*x)^3/b^3+1/3*x^3*arcsinh(a+b*x)^3+14/9*sqrt(1+(a+b*x)^2)/b^3-6*a^2*sqrt(1+(a+b*x)^2)/b^3+3/4*a*(a+b*x)*sqrt(1+(a+b*x)^2)/b^3+2/3*arcsinh(a+b*x)^2*sqrt(1+(a+b*x)^2)/b^3-3*a^2*arcsinh(a+b*x)^2*sqrt(1+(a+b*x)^2)/b^3+3/2*a*(a+b*x)*arcsinh(a+b*x)^2*sqrt(1+(a+b*x)^2)/b^3-1/3*(a+b*x)^2*arcsinh(a+b*x)^2*sqrt(1+(a+b*x)^2)/b^3],
[x*arcsinh(a+b*x)^3,x,12,3/8*arcsinh(a+b*x)/b^2-6*a*(a+b*x)*arcsinh(a+b*x)/b^2+3/4*(a+b*x)^2*arcsinh(a+b*x)/b^2+1/4*arcsinh(a+b*x)^3/b^2-1/2*a^2*arcsinh(a+b*x)^3/b^2+1/2*x^2*arcsinh(a+b*x)^3+6*a*sqrt(1+(a+b*x)^2)/b^2-3/8*(a+b*x)*sqrt(1+(a+b*x)^2)/b^2+3*a*arcsinh(a+b*x)^2*sqrt(1+(a+b*x)^2)/b^2-3/4*(a+b*x)*arcsinh(a+b*x)^2*sqrt(1+(a+b*x)^2)/b^2],
[arcsinh(a+b*x)^3,x,5,6*(a+b*x)*arcsinh(a+b*x)/b+(a+b*x)*arcsinh(a+b*x)^3/b-6*sqrt(1+(a+b*x)^2)/b-3*arcsinh(a+b*x)^2*sqrt(1+(a+b*x)^2)/b],
[arcsinh(a+b*x)^3/x,x,13,-1/4*arcsinh(a+b*x)^4+arcsinh(a+b*x)^3*log(1-exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))+arcsinh(a+b*x)^3*log(1-exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))+3*arcsinh(a+b*x)^2*polylog(2,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))+3*arcsinh(a+b*x)^2*polylog(2,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))-6*arcsinh(a+b*x)*polylog(3,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))-6*arcsinh(a+b*x)*polylog(3,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))+6*polylog(4,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))+6*polylog(4,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))],
[arcsinh(a+b*x)^3/x^2,x,13,-arcsinh(a+b*x)^3/x-3*b*arcsinh(a+b*x)^2*log(1-exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/sqrt(1+a^2)+3*b*arcsinh(a+b*x)^2*log(1-exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/sqrt(1+a^2)-6*b*arcsinh(a+b*x)*polylog(2,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/sqrt(1+a^2)+6*b*arcsinh(a+b*x)*polylog(2,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/sqrt(1+a^2)+6*b*polylog(3,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/sqrt(1+a^2)-6*b*polylog(3,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/sqrt(1+a^2)],
[arcsinh(a+b*x)^3/x^3,x,21,-3/2*b^2*arcsinh(a+b*x)^2/(1+a^2)-1/2*arcsinh(a+b*x)^3/x^2+3*b^2*arcsinh(a+b*x)*log(1-exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/(1+a^2)+3/2*a*b^2*arcsinh(a+b*x)^2*log(1-exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/(1+a^2)^(3/2)+3*b^2*arcsinh(a+b*x)*log(1-exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/(1+a^2)-3/2*a*b^2*arcsinh(a+b*x)^2*log(1-exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/(1+a^2)^(3/2)+3*b^2*polylog(2,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/(1+a^2)+3*a*b^2*arcsinh(a+b*x)*polylog(2,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/(1+a^2)^(3/2)+3*b^2*polylog(2,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/(1+a^2)-3*a*b^2*arcsinh(a+b*x)*polylog(2,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/(1+a^2)^(3/2)-3*a*b^2*polylog(3,exp(arcsinh(a+b*x))/(a-sqrt(1+a^2)))/(1+a^2)^(3/2)+3*a*b^2*polylog(3,exp(arcsinh(a+b*x))/(a+sqrt(1+a^2)))/(1+a^2)^(3/2)-3/2*b*arcsinh(a+b*x)^2*sqrt(1+(a+b*x)^2)/((1+a^2)*x)],

# n<0
[x^2/arcsinh(a+b*x),x,14,-1/4*Chi(arcsinh(a+b*x))/b^3+a^2*Chi(arcsinh(a+b*x))/b^3+1/4*Chi(3*arcsinh(a+b*x))/b^3-a*Shi(2*arcsinh(a+b*x))/b^3],
[x/arcsinh(a+b*x),x,10,-a*Chi(arcsinh(a+b*x))/b^2+1/2*Shi(2*arcsinh(a+b*x))/b^2],
[1/arcsinh(a+b*x),x,3,Chi(arcsinh(a+b*x))/b],
[1/(x*arcsinh(a+b*x)),x,1,Unintegrable(1/(x*arcsinh(a+b*x)),x)],
[x^2/arcsinh(a+b*x)^2,x,12,-2*a*Chi(2*arcsinh(a+b*x))/b^3-1/4*Shi(arcsinh(a+b*x))/b^3+a^2*Shi(arcsinh(a+b*x))/b^3+3/4*Shi(3*arcsinh(a+b*x))/b^3-a^2*sqrt(1+(a+b*x)^2)/(b^3*arcsinh(a+b*x))+2*a*(a+b*x)*sqrt(1+(a+b*x)^2)/(b^3*arcsinh(a+b*x))-(a+b*x)^2*sqrt(1+(a+b*x)^2)/(b^3*arcsinh(a+b*x))],
[x/arcsinh(a+b*x)^2,x,8,Chi(2*arcsinh(a+b*x))/b^2-a*Shi(arcsinh(a+b*x))/b^2+a*sqrt(1+(a+b*x)^2)/(b^2*arcsinh(a+b*x))-(a+b*x)*sqrt(1+(a+b*x)^2)/(b^2*arcsinh(a+b*x))],
[1/arcsinh(a+b*x)^2,x,4,Shi(arcsinh(a+b*x))/b-sqrt(1+(a+b*x)^2)/(b*arcsinh(a+b*x))],
[1/(x*arcsinh(a+b*x)^2),x,1,Unintegrable(1/(x*arcsinh(a+b*x)^2),x)],
[x^2/arcsinh(a+b*x)^3,x,24,a/(b^3*arcsinh(a+b*x))+(-a-b*x)/(b^3*arcsinh(a+b*x))-1/2*a^2*(a+b*x)/(b^3*arcsinh(a+b*x))+2*a*(a+b*x)^2/(b^3*arcsinh(a+b*x))-3/2*(a+b*x)^3/(b^3*arcsinh(a+b*x))-1/8*Chi(arcsinh(a+b*x))/b^3+1/2*a^2*Chi(arcsinh(a+b*x))/b^3+9/8*Chi(3*arcsinh(a+b*x))/b^3-2*a*Shi(2*arcsinh(a+b*x))/b^3-1/2*a^2*sqrt(1+(a+b*x)^2)/(b^3*arcsinh(a+b*x)^2)+a*(a+b*x)*sqrt(1+(a+b*x)^2)/(b^3*arcsinh(a+b*x)^2)-1/2*(a+b*x)^2*sqrt(1+(a+b*x)^2)/(b^3*arcsinh(a+b*x)^2)],
[x/arcsinh(a+b*x)^3,x,14,(-1/2)/(b^2*arcsinh(a+b*x))+1/2*a*(a+b*x)/(b^2*arcsinh(a+b*x))-(a+b*x)^2/(b^2*arcsinh(a+b*x))-1/2*a*Chi(arcsinh(a+b*x))/b^2+Shi(2*arcsinh(a+b*x))/b^2+1/2*a*sqrt(1+(a+b*x)^2)/(b^2*arcsinh(a+b*x)^2)-1/2*(a+b*x)*sqrt(1+(a+b*x)^2)/(b^2*arcsinh(a+b*x)^2)],
[1/arcsinh(a+b*x)^3,x,5,1/2*(-a-b*x)/(b*arcsinh(a+b*x))+1/2*Chi(arcsinh(a+b*x))/b-1/2*sqrt(1+(a+b*x)^2)/(b*arcsinh(a+b*x)^2)],
[1/(x*arcsinh(a+b*x)^3),x,1,Unintegrable(1/(x*arcsinh(a+b*x)^3),x)],

# n symbolic
[x^m*(a+b*arcsinh(c+d*x))^n,x,1,Unintegrable(x^m*(a+b*arcsinh(c+d*x))^n,x)],
[x^2*(a+b*arcsinh(c+d*x))^n,x,22,1/8*3^(-1-n)*(a+b*arcsinh(c+d*x))^n*GAMMA(1+n,-3*(a+b*arcsinh(c+d*x))/b)/(d^3*exp(3*a/b)*((-a-b*arcsinh(c+d*x))/b)^n)-2^(-2-n)*c*(a+b*arcsinh(c+d*x))^n*GAMMA(1+n,-2*(a+b*arcsinh(c+d*x))/b)/(d^3*exp(2*a/b)*((-a-b*arcsinh(c+d*x))/b)^n)-1/8*(a+b*arcsinh(c+d*x))^n*GAMMA(1+n,(-a-b*arcsinh(c+d*x))/b)/(d^3*exp(a/b)*((-a-b*arcsinh(c+d*x))/b)^n)+1/2*c^2*(a+b*arcsinh(c+d*x))^n*GAMMA(1+n,(-a-b*arcsinh(c+d*x))/b)/(d^3*exp(a/b)*((-a-b*arcsinh(c+d*x))/b)^n)+1/8*exp(a/b)*(a+b*arcsinh(c+d*x))^n*GAMMA(1+n,(a+b*arcsinh(c+d*x))/b)/(d^3*((a+b*arcsinh(c+d*x))/b)^n)-1/2*c^2*exp(a/b)*(a+b*arcsinh(c+d*x))^n*GAMMA(1+n,(a+b*arcsinh(c+d*x))/b)/(d^3*((a+b*arcsinh(c+d*x))/b)^n)-2^(-2-n)*c*exp(2*a/b)*(a+b*arcsinh(c+d*x))^n*GAMMA(1+n,2*(a+b*arcsinh(c+d*x))/b)/(d^3*((a+b*arcsinh(c+d*x))/b)^n)-1/8*3^(-1-n)*exp(3*a/b)*(a+b*arcsinh(c+d*x))^n*GAMMA(1+n,3*(a+b*arcsinh(c+d*x))/b)/(d^3*((a+b*arcsinh(c+d*x))/b)^n)],
[x*(a+b*arcsinh(c+d*x))^n,x,14,2^(-3-n)*(a+b*arcsinh(c+d*x))^n*GAMMA(1+n,-2*(a+b*arcsinh(c+d*x))/b)/(d^2*exp(2*a/b)*((-a-b*arcsinh(c+d*x))/b)^n)-1/2*c*(a+b*arcsinh(c+d*x))^n*GAMMA(1+n,(-a-b*arcsinh(c+d*x))/b)/(d^2*exp(a/b)*((-a-b*arcsinh(c+d*x))/b)^n)+1/2*c*exp(a/b)*(a+b*arcsinh(c+d*x))^n*GAMMA(1+n,(a+b*arcsinh(c+d*x))/b)/(d^2*((a+b*arcsinh(c+d*x))/b)^n)+2^(-3-n)*exp(2*a/b)*(a+b*arcsinh(c+d*x))^n*GAMMA(1+n,2*(a+b*arcsinh(c+d*x))/b)/(d^2*((a+b*arcsinh(c+d*x))/b)^n)],
[(a+b*arcsinh(c+d*x))^n,x,5,1/2*(a+b*arcsinh(c+d*x))^n*GAMMA(1+n,(-a-b*arcsinh(c+d*x))/b)/(d*exp(a/b)*((-a-b*arcsinh(c+d*x))/b)^n)-1/2*exp(a/b)*(a+b*arcsinh(c+d*x))^n*GAMMA(1+n,(a+b*arcsinh(c+d*x))/b)/(d*((a+b*arcsinh(c+d*x))/b)^n)],
[(a+b*arcsinh(c+d*x))^n/x,x,1,Unintegrable((a+b*arcsinh(c+d*x))^n/x,x)],

# Integrands of the form x^m (a+b ArcSinh[c+d x])^(n/2)

# n>0
[x^2*(a+b*arcsinh(c+d*x))^(1/2),x,23,1/48*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/3*Pi)/d^3-1/48*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/3*Pi)/(d^3*exp(3*a/b))+1/8*c*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/d^3+1/8*c*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/(d^3*exp(2*a/b))-1/16*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/d^3+1/4*c^2*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/d^3+1/16*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/(d^3*exp(a/b))-1/4*c^2*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/(d^3*exp(a/b))+c^2*(c+d*x)*sqrt(a+b*arcsinh(c+d*x))/d^3+1/3*(c+d*x)^3*sqrt(a+b*arcsinh(c+d*x))/d^3-1/2*c*cosh(2*arcsinh(c+d*x))*sqrt(a+b*arcsinh(c+d*x))/d^3],
[x*(a+b*arcsinh(c+d*x))^(1/2),x,14,-1/16*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/d^2-1/16*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/(d^2*exp(2*a/b))-1/4*c*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/d^2+1/4*c*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/(d^2*exp(a/b))-c*(c+d*x)*sqrt(a+b*arcsinh(c+d*x))/d^2+1/4*cosh(2*arcsinh(c+d*x))*sqrt(a+b*arcsinh(c+d*x))/d^2],
[(a+b*arcsinh(c+d*x))^(1/2),x,8,1/4*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/d-1/4*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/(d*exp(a/b))+(c+d*x)*sqrt(a+b*arcsinh(c+d*x))/d],
[x*(a+b*arcsinh(c+d*x))^(3/2),x,16,-c*(c+d*x)*(a+b*arcsinh(c+d*x))^(3/2)/d^2+1/4*(a+b*arcsinh(c+d*x))^(3/2)*cosh(2*arcsinh(c+d*x))/d^2-3/64*b^(3/2)*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d^2+3/64*b^(3/2)*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d^2*exp(2*a/b))-3/8*b^(3/2)*c*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d^2-3/8*b^(3/2)*c*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d^2*exp(a/b))-3/16*b*sinh(2*arcsinh(c+d*x))*sqrt(a+b*arcsinh(c+d*x))/d^2+3/2*b*c*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d^2],
[(a+b*arcsinh(c+d*x))^(3/2),x,9,(c+d*x)*(a+b*arcsinh(c+d*x))^(3/2)/d+3/8*b^(3/2)*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d+3/8*b^(3/2)*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b))-3/2*b*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d],
[x*(a+b*arcsinh(c+d*x))^(5/2),x,18,-c*(c+d*x)*(a+b*arcsinh(c+d*x))^(5/2)/d^2+1/4*(a+b*arcsinh(c+d*x))^(5/2)*cosh(2*arcsinh(c+d*x))/d^2-5/16*b*(a+b*arcsinh(c+d*x))^(3/2)*sinh(2*arcsinh(c+d*x))/d^2-15/256*b^(5/2)*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d^2-15/256*b^(5/2)*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d^2*exp(2*a/b))-15/16*b^(5/2)*c*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d^2+15/16*b^(5/2)*c*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d^2*exp(a/b))+5/2*b*c*(a+b*arcsinh(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)/d^2-15/4*b^2*c*(c+d*x)*sqrt(a+b*arcsinh(c+d*x))/d^2+15/64*b^2*cosh(2*arcsinh(c+d*x))*sqrt(a+b*arcsinh(c+d*x))/d^2],
[(a+b*arcsinh(c+d*x))^(5/2),x,10,(c+d*x)*(a+b*arcsinh(c+d*x))^(5/2)/d+15/16*b^(5/2)*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d-15/16*b^(5/2)*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b))-5/2*b*(a+b*arcsinh(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)/d+15/4*b^2*(c+d*x)*sqrt(a+b*arcsinh(c+d*x))/d],

# n<0
[x^2/(a+b*arcsinh(c+d*x))^(1/2),x,20,1/8*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/(d^3*sqrt(b))+1/8*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/(d^3*exp(3*a/b)*sqrt(b))+1/2*c*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d^3*sqrt(b))-1/2*c*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d^3*exp(2*a/b)*sqrt(b))-1/8*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d^3*sqrt(b))+1/2*c^2*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d^3*sqrt(b))-1/8*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d^3*exp(a/b)*sqrt(b))+1/2*c^2*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d^3*exp(a/b)*sqrt(b))],
[x/(a+b*arcsinh(c+d*x))^(1/2),x,12,-1/4*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d^2*sqrt(b))+1/4*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d^2*exp(2*a/b)*sqrt(b))-1/2*c*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d^2*sqrt(b))-1/2*c*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d^2*exp(a/b)*sqrt(b))],
[1/(a+b*arcsinh(c+d*x))^(1/2),x,7,1/2*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*sqrt(b))+1/2*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b)*sqrt(b))],
[x/(a+b*arcsinh(c+d*x))^(3/2),x,16,exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(b^(3/2)*d^2)+erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(b^(3/2)*d^2*exp(2*a/b))+c*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*d^2)-c*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*d^2*exp(a/b))+2*c*sqrt(1+(c+d*x)^2)/(b*d^2*sqrt(a+b*arcsinh(c+d*x)))-2*(c+d*x)*sqrt(1+(c+d*x)^2)/(b*d^2*sqrt(a+b*arcsinh(c+d*x)))],
[1/(a+b*arcsinh(c+d*x))^(3/2),x,8,-exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*d)+erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*d*exp(a/b))-2*sqrt(1+(c+d*x)^2)/(b*d*sqrt(a+b*arcsinh(c+d*x)))],
[x/(a+b*arcsinh(c+d*x))^(5/2),x,22,-2/3*c*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*d^2)-2/3*c*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*d^2*exp(a/b))-2/3*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(5/2)*d^2)+2/3*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(5/2)*d^2*exp(2*a/b))+2/3*c*sqrt(1+(c+d*x)^2)/(b*d^2*(a+b*arcsinh(c+d*x))^(3/2))-2/3*(c+d*x)*sqrt(1+(c+d*x)^2)/(b*d^2*(a+b*arcsinh(c+d*x))^(3/2))+(-4/3)/(b^2*d^2*sqrt(a+b*arcsinh(c+d*x)))+4/3*c*(c+d*x)/(b^2*d^2*sqrt(a+b*arcsinh(c+d*x)))-8/3*(c+d*x)^2/(b^2*d^2*sqrt(a+b*arcsinh(c+d*x)))],
[1/(a+b*arcsinh(c+d*x))^(5/2),x,9,2/3*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*d)+2/3*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*d*exp(a/b))-2/3*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^(3/2))-4/3*(c+d*x)/(b^2*d*sqrt(a+b*arcsinh(c+d*x)))],
[x/(a+b*arcsinh(c+d*x))^(7/2),x,21,(-4/15)/(b^2*d^2*(a+b*arcsinh(c+d*x))^(3/2))+4/15*c*(c+d*x)/(b^2*d^2*(a+b*arcsinh(c+d*x))^(3/2))-8/15*(c+d*x)^2/(b^2*d^2*(a+b*arcsinh(c+d*x))^(3/2))+4/15*c*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*d^2)-4/15*c*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*d^2*exp(a/b))+8/15*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(7/2)*d^2)+8/15*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(7/2)*d^2*exp(2*a/b))+2/5*c*sqrt(1+(c+d*x)^2)/(b*d^2*(a+b*arcsinh(c+d*x))^(5/2))-2/5*(c+d*x)*sqrt(1+(c+d*x)^2)/(b*d^2*(a+b*arcsinh(c+d*x))^(5/2))+8/15*c*sqrt(1+(c+d*x)^2)/(b^3*d^2*sqrt(a+b*arcsinh(c+d*x)))-32/15*(c+d*x)*sqrt(1+(c+d*x)^2)/(b^3*d^2*sqrt(a+b*arcsinh(c+d*x)))],
[1/(a+b*arcsinh(c+d*x))^(7/2),x,10,-4/15*(c+d*x)/(b^2*d*(a+b*arcsinh(c+d*x))^(3/2))-4/15*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*d)+4/15*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*d*exp(a/b))-2/5*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^(5/2))-8/15*sqrt(1+(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsinh(c+d*x)))],

# Integrands of the form (c e+d e x)^m (a+b ArcSinh[c+d x])^n

# n>0
[(c*e+d*e*x)^m*(a+b*arcsinh(c+d*x)),x,3,(e*(c+d*x))^(1+m)*(a+b*arcsinh(c+d*x))/(d*e*(1+m))-b*(e*(c+d*x))^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],-(c+d*x)^2)/(d*e^2*(1+m)*(2+m))],
[(c*e+d*e*x)^4*(a+b*arcsinh(c+d*x)),x,6,2/15*b*e^4*(1+(c+d*x)^2)^(3/2)/d-1/25*b*e^4*(1+(c+d*x)^2)^(5/2)/d+1/5*e^4*(c+d*x)^5*(a+b*arcsinh(c+d*x))/d-1/5*b*e^4*sqrt(1+(c+d*x)^2)/d],
[(c*e+d*e*x)^3*(a+b*arcsinh(c+d*x)),x,6,-3/32*b*e^3*arcsinh(c+d*x)/d+1/4*e^3*(c+d*x)^4*(a+b*arcsinh(c+d*x))/d+3/32*b*e^3*(c+d*x)*sqrt(1+(c+d*x)^2)/d-1/16*b*e^3*(c+d*x)^3*sqrt(1+(c+d*x)^2)/d],
[(c*e+d*e*x)^2*(a+b*arcsinh(c+d*x)),x,6,-1/9*b*e^2*(1+(c+d*x)^2)^(3/2)/d+1/3*e^2*(c+d*x)^3*(a+b*arcsinh(c+d*x))/d+1/3*b*e^2*sqrt(1+(c+d*x)^2)/d],
[(c*e+d*e*x)*(a+b*arcsinh(c+d*x)),x,5,1/4*b*e*arcsinh(c+d*x)/d+1/2*e*(c+d*x)^2*(a+b*arcsinh(c+d*x))/d-1/4*b*e*(c+d*x)*sqrt(1+(c+d*x)^2)/d],
[a+b*arcsinh(c+d*x),x,4,a*x+b*(c+d*x)*arcsinh(c+d*x)/d-b*sqrt(1+(c+d*x)^2)/d],
[(a+b*arcsinh(c+d*x))/(c*e+d*e*x),x,7,1/2*(a+b*arcsinh(c+d*x))^2/(b*d*e)+(a+b*arcsinh(c+d*x))*log(1+(-1)/exp(2*arcsinh(c+d*x)))/(d*e)-1/2*b*polylog(2,exp(-2*arcsinh(c+d*x)))/(d*e)],
[(a+b*arcsinh(c+d*x))/(c*e+d*e*x)^2,x,6,(-a-b*arcsinh(c+d*x))/(d*e^2*(c+d*x))-b*arctanh(sqrt(1+(c+d*x)^2))/(d*e^2)],
[(a+b*arcsinh(c+d*x))/(c*e+d*e*x)^3,x,4,1/2*(-a-b*arcsinh(c+d*x))/(d*e^3*(c+d*x)^2)-1/2*b*sqrt(1+(c+d*x)^2)/(d*e^3*(c+d*x))],
[(a+b*arcsinh(c+d*x))/(c*e+d*e*x)^4,x,7,1/3*(-a-b*arcsinh(c+d*x))/(d*e^4*(c+d*x)^3)+1/6*b*arctanh(sqrt(1+(c+d*x)^2))/(d*e^4)-1/6*b*sqrt(1+(c+d*x)^2)/(d*e^4*(c+d*x)^2)],
[(a+b*arcsinh(c+d*x))/(c*e+d*e*x)^5,x,5,1/4*(-a-b*arcsinh(c+d*x))/(d*e^5*(c+d*x)^4)-1/12*b*sqrt(1+(c+d*x)^2)/(d*e^5*(c+d*x)^3)+1/6*b*sqrt(1+(c+d*x)^2)/(d*e^5*(c+d*x))],
[(a+b*arcsinh(c+d*x))/(c*e+d*e*x)^6,x,8,1/5*(-a-b*arcsinh(c+d*x))/(d*e^6*(c+d*x)^5)-3/40*b*arctanh(sqrt(1+(c+d*x)^2))/(d*e^6)-1/20*b*sqrt(1+(c+d*x)^2)/(d*e^6*(c+d*x)^4)+3/40*b*sqrt(1+(c+d*x)^2)/(d*e^6*(c+d*x)^2)],
[(c*e+d*e*x)^m*(a+b*arcsinh(c+d*x))^2,x,3,(e*(c+d*x))^(1+m)*(a+b*arcsinh(c+d*x))^2/(d*e*(1+m))-2*b*(e*(c+d*x))^(2+m)*(a+b*arcsinh(c+d*x))*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],-(c+d*x)^2)/(d*e^2*(1+m)*(2+m))+2*b^2*(e*(c+d*x))^(3+m)*HypergeometricPFQ([1,3/2+1/2*m,3/2+1/2*m],[2+1/2*m,5/2+1/2*m],-(c+d*x)^2)/(d*e^3*(3+m)*(2+3*m+m^2))],
[(c*e+d*e*x)^4*(a+b*arcsinh(c+d*x))^2,x,9,16/75*b^2*e^4*x-8/225*b^2*e^4*(c+d*x)^3/d+2/125*b^2*e^4*(c+d*x)^5/d+1/5*e^4*(c+d*x)^5*(a+b*arcsinh(c+d*x))^2/d-16/75*b*e^4*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d+8/75*b*e^4*(c+d*x)^2*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d-2/25*b*e^4*(c+d*x)^4*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d],
[(c*e+d*e*x)^3*(a+b*arcsinh(c+d*x))^2,x,8,-3/32*b^2*e^3*(c+d*x)^2/d+1/32*b^2*e^3*(c+d*x)^4/d-3/32*e^3*(a+b*arcsinh(c+d*x))^2/d+1/4*e^3*(c+d*x)^4*(a+b*arcsinh(c+d*x))^2/d+3/16*b*e^3*(c+d*x)*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d-1/8*b*e^3*(c+d*x)^3*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d],
[(c*e+d*e*x)^2*(a+b*arcsinh(c+d*x))^2,x,7,-4/9*b^2*e^2*x+2/27*b^2*e^2*(c+d*x)^3/d+1/3*e^2*(c+d*x)^3*(a+b*arcsinh(c+d*x))^2/d+4/9*b*e^2*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d-2/9*b*e^2*(c+d*x)^2*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d],
[(c*e+d*e*x)*(a+b*arcsinh(c+d*x))^2,x,6,1/4*b^2*e*(c+d*x)^2/d+1/4*e*(a+b*arcsinh(c+d*x))^2/d+1/2*e*(c+d*x)^2*(a+b*arcsinh(c+d*x))^2/d-1/2*b*e*(c+d*x)*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d],
[(a+b*arcsinh(c+d*x))^2,x,4,2*b^2*x+(c+d*x)*(a+b*arcsinh(c+d*x))^2/d-2*b*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d],
[(a+b*arcsinh(c+d*x))^2/(c*e+d*e*x),x,8,1/3*(a+b*arcsinh(c+d*x))^3/(b*d*e)+(a+b*arcsinh(c+d*x))^2*log(1+(-1)/exp(2*arcsinh(c+d*x)))/(d*e)-b*(a+b*arcsinh(c+d*x))*polylog(2,exp(-2*arcsinh(c+d*x)))/(d*e)-1/2*b^2*polylog(3,exp(-2*arcsinh(c+d*x)))/(d*e)],
[(a+b*arcsinh(c+d*x))^2/(c*e+d*e*x)^2,x,9,-(a+b*arcsinh(c+d*x))^2/(d*e^2*(c+d*x))-4*b*(a+b*arcsinh(c+d*x))*arctanh(exp(arcsinh(c+d*x)))/(d*e^2)-2*b^2*polylog(2,-exp(arcsinh(c+d*x)))/(d*e^2)+2*b^2*polylog(2,exp(arcsinh(c+d*x)))/(d*e^2)],
[(a+b*arcsinh(c+d*x))^2/(c*e+d*e*x)^3,x,5,-1/2*(a+b*arcsinh(c+d*x))^2/(d*e^3*(c+d*x)^2)+b^2*log(c+d*x)/(d*e^3)-b*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/(d*e^3*(c+d*x))],
[(a+b*arcsinh(c+d*x))^2/(c*e+d*e*x)^4,x,11,-1/3*b^2/(d*e^4*(c+d*x))-1/3*(a+b*arcsinh(c+d*x))^2/(d*e^4*(c+d*x)^3)+2/3*b*(a+b*arcsinh(c+d*x))*arctanh(exp(arcsinh(c+d*x)))/(d*e^4)+1/3*b^2*polylog(2,-exp(arcsinh(c+d*x)))/(d*e^4)-1/3*b^2*polylog(2,exp(arcsinh(c+d*x)))/(d*e^4)-1/3*b*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/(d*e^4*(c+d*x)^2)],
[(c*e+d*e*x)^m*(a+b*arcsinh(c+d*x))^3,x,2,(e*(c+d*x))^(1+m)*(a+b*arcsinh(c+d*x))^3/(d*e*(1+m))-3*b*Unintegrable((e*(c+d*x))^(1+m)*(a+b*arcsinh(c+d*x))^2/sqrt(1+(c+d*x)^2),x)/(e*(1+m))],
[(c*e+d*e*x)^4*(a+b*arcsinh(c+d*x))^3,x,17,16/25*a*b^2*e^4*x+76/1125*b^3*e^4*(1+(c+d*x)^2)^(3/2)/d-6/625*b^3*e^4*(1+(c+d*x)^2)^(5/2)/d+16/25*b^3*e^4*(c+d*x)*arcsinh(c+d*x)/d-8/75*b^2*e^4*(c+d*x)^3*(a+b*arcsinh(c+d*x))/d+6/125*b^2*e^4*(c+d*x)^5*(a+b*arcsinh(c+d*x))/d+1/5*e^4*(c+d*x)^5*(a+b*arcsinh(c+d*x))^3/d-298/375*b^3*e^4*sqrt(1+(c+d*x)^2)/d-8/25*b*e^4*(a+b*arcsinh(c+d*x))^2*sqrt(1+(c+d*x)^2)/d+4/25*b*e^4*(c+d*x)^2*(a+b*arcsinh(c+d*x))^2*sqrt(1+(c+d*x)^2)/d-3/25*b*e^4*(c+d*x)^4*(a+b*arcsinh(c+d*x))^2*sqrt(1+(c+d*x)^2)/d],
[(c*e+d*e*x)^3*(a+b*arcsinh(c+d*x))^3,x,13,-45/256*b^3*e^3*arcsinh(c+d*x)/d-9/32*b^2*e^3*(c+d*x)^2*(a+b*arcsinh(c+d*x))/d+3/32*b^2*e^3*(c+d*x)^4*(a+b*arcsinh(c+d*x))/d-3/32*e^3*(a+b*arcsinh(c+d*x))^3/d+1/4*e^3*(c+d*x)^4*(a+b*arcsinh(c+d*x))^3/d+45/256*b^3*e^3*(c+d*x)*sqrt(1+(c+d*x)^2)/d-3/128*b^3*e^3*(c+d*x)^3*sqrt(1+(c+d*x)^2)/d+9/32*b*e^3*(c+d*x)*(a+b*arcsinh(c+d*x))^2*sqrt(1+(c+d*x)^2)/d-3/16*b*e^3*(c+d*x)^3*(a+b*arcsinh(c+d*x))^2*sqrt(1+(c+d*x)^2)/d],
[(c*e+d*e*x)^2*(a+b*arcsinh(c+d*x))^3,x,12,-4/3*a*b^2*e^2*x-2/27*b^3*e^2*(1+(c+d*x)^2)^(3/2)/d-4/3*b^3*e^2*(c+d*x)*arcsinh(c+d*x)/d+2/9*b^2*e^2*(c+d*x)^3*(a+b*arcsinh(c+d*x))/d+1/3*e^2*(c+d*x)^3*(a+b*arcsinh(c+d*x))^3/d+14/9*b^3*e^2*sqrt(1+(c+d*x)^2)/d+2/3*b*e^2*(a+b*arcsinh(c+d*x))^2*sqrt(1+(c+d*x)^2)/d-1/3*b*e^2*(c+d*x)^2*(a+b*arcsinh(c+d*x))^2*sqrt(1+(c+d*x)^2)/d],
[(c*e+d*e*x)*(a+b*arcsinh(c+d*x))^3,x,8,3/8*b^3*e*arcsinh(c+d*x)/d+3/4*b^2*e*(c+d*x)^2*(a+b*arcsinh(c+d*x))/d+1/4*e*(a+b*arcsinh(c+d*x))^3/d+1/2*e*(c+d*x)^2*(a+b*arcsinh(c+d*x))^3/d-3/8*b^3*e*(c+d*x)*sqrt(1+(c+d*x)^2)/d-3/4*b*e*(c+d*x)*(a+b*arcsinh(c+d*x))^2*sqrt(1+(c+d*x)^2)/d],
[(a+b*arcsinh(c+d*x))^3,x,6,6*a*b^2*x+6*b^3*(c+d*x)*arcsinh(c+d*x)/d+(c+d*x)*(a+b*arcsinh(c+d*x))^3/d-6*b^3*sqrt(1+(c+d*x)^2)/d-3*b*(a+b*arcsinh(c+d*x))^2*sqrt(1+(c+d*x)^2)/d],
[(a+b*arcsinh(c+d*x))^3/(c*e+d*e*x),x,9,1/4*(a+b*arcsinh(c+d*x))^4/(b*d*e)+(a+b*arcsinh(c+d*x))^3*log(1+(-1)/exp(2*arcsinh(c+d*x)))/(d*e)-3/2*b*(a+b*arcsinh(c+d*x))^2*polylog(2,exp(-2*arcsinh(c+d*x)))/(d*e)-3/2*b^2*(a+b*arcsinh(c+d*x))*polylog(3,exp(-2*arcsinh(c+d*x)))/(d*e)-3/4*b^3*polylog(4,exp(-2*arcsinh(c+d*x)))/(d*e)],
[(a+b*arcsinh(c+d*x))^3/(c*e+d*e*x)^2,x,11,-(a+b*arcsinh(c+d*x))^3/(d*e^2*(c+d*x))-6*b*(a+b*arcsinh(c+d*x))^2*arctanh(exp(arcsinh(c+d*x)))/(d*e^2)-6*b^2*(a+b*arcsinh(c+d*x))*polylog(2,-exp(arcsinh(c+d*x)))/(d*e^2)+6*b^2*(a+b*arcsinh(c+d*x))*polylog(2,exp(arcsinh(c+d*x)))/(d*e^2)+6*b^3*polylog(3,-exp(arcsinh(c+d*x)))/(d*e^2)-6*b^3*polylog(3,exp(arcsinh(c+d*x)))/(d*e^2)],
[(a+b*arcsinh(c+d*x))^3/(c*e+d*e*x)^3,x,9,3/2*b*(a+b*arcsinh(c+d*x))^2/(d*e^3)-1/2*(a+b*arcsinh(c+d*x))^3/(d*e^3*(c+d*x)^2)+3*b^2*(a+b*arcsinh(c+d*x))*log(1+(-1)/exp(2*arcsinh(c+d*x)))/(d*e^3)-3/2*b^3*polylog(2,exp(-2*arcsinh(c+d*x)))/(d*e^3)-3/2*b*(a+b*arcsinh(c+d*x))^2*sqrt(1+(c+d*x)^2)/(d*e^3*(c+d*x))],
[(a+b*arcsinh(c+d*x))^3/(c*e+d*e*x)^4,x,16,-b^2*(a+b*arcsinh(c+d*x))/(d*e^4*(c+d*x))-1/3*(a+b*arcsinh(c+d*x))^3/(d*e^4*(c+d*x)^3)+b*(a+b*arcsinh(c+d*x))^2*arctanh(exp(arcsinh(c+d*x)))/(d*e^4)-b^3*arctanh(sqrt(1+(c+d*x)^2))/(d*e^4)+b^2*(a+b*arcsinh(c+d*x))*polylog(2,-exp(arcsinh(c+d*x)))/(d*e^4)-b^2*(a+b*arcsinh(c+d*x))*polylog(2,exp(arcsinh(c+d*x)))/(d*e^4)-b^3*polylog(3,-exp(arcsinh(c+d*x)))/(d*e^4)+b^3*polylog(3,exp(arcsinh(c+d*x)))/(d*e^4)-1/2*b*(a+b*arcsinh(c+d*x))^2*sqrt(1+(c+d*x)^2)/(d*e^4*(c+d*x)^2)],
[(c*e+d*e*x)^m*(a+b*arcsinh(c+d*x))^4,x,2,(e*(c+d*x))^(1+m)*(a+b*arcsinh(c+d*x))^4/(d*e*(1+m))-4*b*Unintegrable((e*(c+d*x))^(1+m)*(a+b*arcsinh(c+d*x))^3/sqrt(1+(c+d*x)^2),x)/(e*(1+m))],
[(c*e+d*e*x)^3*(a+b*arcsinh(c+d*x))^4,x,16,-45/128*b^4*e^3*(c+d*x)^2/d+3/128*b^4*e^3*(c+d*x)^4/d-45/128*b^2*e^3*(a+b*arcsinh(c+d*x))^2/d-9/16*b^2*e^3*(c+d*x)^2*(a+b*arcsinh(c+d*x))^2/d+3/16*b^2*e^3*(c+d*x)^4*(a+b*arcsinh(c+d*x))^2/d-3/32*e^3*(a+b*arcsinh(c+d*x))^4/d+1/4*e^3*(c+d*x)^4*(a+b*arcsinh(c+d*x))^4/d+45/64*b^3*e^3*(c+d*x)*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d-3/32*b^3*e^3*(c+d*x)^3*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d+3/8*b*e^3*(c+d*x)*(a+b*arcsinh(c+d*x))^3*sqrt(1+(c+d*x)^2)/d-1/4*b*e^3*(c+d*x)^3*(a+b*arcsinh(c+d*x))^3*sqrt(1+(c+d*x)^2)/d],
[(c*e+d*e*x)^2*(a+b*arcsinh(c+d*x))^4,x,13,-160/27*b^4*e^2*x+8/81*b^4*e^2*(c+d*x)^3/d-8/3*b^2*e^2*(c+d*x)*(a+b*arcsinh(c+d*x))^2/d+4/9*b^2*e^2*(c+d*x)^3*(a+b*arcsinh(c+d*x))^2/d+1/3*e^2*(c+d*x)^3*(a+b*arcsinh(c+d*x))^4/d+160/27*b^3*e^2*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d-8/27*b^3*e^2*(c+d*x)^2*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d+8/9*b*e^2*(a+b*arcsinh(c+d*x))^3*sqrt(1+(c+d*x)^2)/d-4/9*b*e^2*(c+d*x)^2*(a+b*arcsinh(c+d*x))^3*sqrt(1+(c+d*x)^2)/d],
[(c*e+d*e*x)*(a+b*arcsinh(c+d*x))^4,x,9,3/4*b^4*e*(c+d*x)^2/d+3/4*b^2*e*(a+b*arcsinh(c+d*x))^2/d+3/2*b^2*e*(c+d*x)^2*(a+b*arcsinh(c+d*x))^2/d+1/4*e*(a+b*arcsinh(c+d*x))^4/d+1/2*e*(c+d*x)^2*(a+b*arcsinh(c+d*x))^4/d-3/2*b^3*e*(c+d*x)*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d-b*e*(c+d*x)*(a+b*arcsinh(c+d*x))^3*sqrt(1+(c+d*x)^2)/d],
[(a+b*arcsinh(c+d*x))^4,x,6,24*b^4*x+12*b^2*(c+d*x)*(a+b*arcsinh(c+d*x))^2/d+(c+d*x)*(a+b*arcsinh(c+d*x))^4/d-24*b^3*(a+b*arcsinh(c+d*x))*sqrt(1+(c+d*x)^2)/d-4*b*(a+b*arcsinh(c+d*x))^3*sqrt(1+(c+d*x)^2)/d],
[(a+b*arcsinh(c+d*x))^4/(c*e+d*e*x),x,10,1/5*(a+b*arcsinh(c+d*x))^5/(b*d*e)+(a+b*arcsinh(c+d*x))^4*log(1+(-1)/exp(2*arcsinh(c+d*x)))/(d*e)-2*b*(a+b*arcsinh(c+d*x))^3*polylog(2,exp(-2*arcsinh(c+d*x)))/(d*e)-3*b^2*(a+b*arcsinh(c+d*x))^2*polylog(3,exp(-2*arcsinh(c+d*x)))/(d*e)-3*b^3*(a+b*arcsinh(c+d*x))*polylog(4,exp(-2*arcsinh(c+d*x)))/(d*e)-3/2*b^4*polylog(5,exp(-2*arcsinh(c+d*x)))/(d*e)],
[(a+b*arcsinh(c+d*x))^4/(c*e+d*e*x)^2,x,13,-(a+b*arcsinh(c+d*x))^4/(d*e^2*(c+d*x))-8*b*(a+b*arcsinh(c+d*x))^3*arctanh(exp(arcsinh(c+d*x)))/(d*e^2)-12*b^2*(a+b*arcsinh(c+d*x))^2*polylog(2,-exp(arcsinh(c+d*x)))/(d*e^2)+12*b^2*(a+b*arcsinh(c+d*x))^2*polylog(2,exp(arcsinh(c+d*x)))/(d*e^2)+24*b^3*(a+b*arcsinh(c+d*x))*polylog(3,-exp(arcsinh(c+d*x)))/(d*e^2)-24*b^3*(a+b*arcsinh(c+d*x))*polylog(3,exp(arcsinh(c+d*x)))/(d*e^2)-24*b^4*polylog(4,-exp(arcsinh(c+d*x)))/(d*e^2)+24*b^4*polylog(4,exp(arcsinh(c+d*x)))/(d*e^2)],
[(a+b*arcsinh(c+d*x))^4/(c*e+d*e*x)^3,x,10,2*b*(a+b*arcsinh(c+d*x))^3/(d*e^3)-1/2*(a+b*arcsinh(c+d*x))^4/(d*e^3*(c+d*x)^2)+6*b^2*(a+b*arcsinh(c+d*x))^2*log(1+(-1)/exp(2*arcsinh(c+d*x)))/(d*e^3)-6*b^3*(a+b*arcsinh(c+d*x))*polylog(2,exp(-2*arcsinh(c+d*x)))/(d*e^3)-3*b^4*polylog(3,exp(-2*arcsinh(c+d*x)))/(d*e^3)-2*b*(a+b*arcsinh(c+d*x))^3*sqrt(1+(c+d*x)^2)/(d*e^3*(c+d*x))],
[(a+b*arcsinh(c+d*x))^4/(c*e+d*e*x)^4,x,21,-2*b^2*(a+b*arcsinh(c+d*x))^2/(d*e^4*(c+d*x))-1/3*(a+b*arcsinh(c+d*x))^4/(d*e^4*(c+d*x)^3)-8*b^3*(a+b*arcsinh(c+d*x))*arctanh(exp(arcsinh(c+d*x)))/(d*e^4)+4/3*b*(a+b*arcsinh(c+d*x))^3*arctanh(exp(arcsinh(c+d*x)))/(d*e^4)-4*b^4*polylog(2,-exp(arcsinh(c+d*x)))/(d*e^4)+2*b^2*(a+b*arcsinh(c+d*x))^2*polylog(2,-exp(arcsinh(c+d*x)))/(d*e^4)+4*b^4*polylog(2,exp(arcsinh(c+d*x)))/(d*e^4)-2*b^2*(a+b*arcsinh(c+d*x))^2*polylog(2,exp(arcsinh(c+d*x)))/(d*e^4)-4*b^3*(a+b*arcsinh(c+d*x))*polylog(3,-exp(arcsinh(c+d*x)))/(d*e^4)+4*b^3*(a+b*arcsinh(c+d*x))*polylog(3,exp(arcsinh(c+d*x)))/(d*e^4)+4*b^4*polylog(4,-exp(arcsinh(c+d*x)))/(d*e^4)-4*b^4*polylog(4,exp(arcsinh(c+d*x)))/(d*e^4)-2/3*b*(a+b*arcsinh(c+d*x))^3*sqrt(1+(c+d*x)^2)/(d*e^4*(c+d*x)^2)],

# n<0
[(c*e+d*e*x)^m/(a+b*arcsinh(c+d*x)),x,1,Unintegrable((e*(c+d*x))^m/(a+b*arcsinh(c+d*x)),x)],
[(c*e+d*e*x)^4/(a+b*arcsinh(c+d*x)),x,14,1/8*e^4*Chi((a+b*arcsinh(c+d*x))/b)*cosh(a/b)/(b*d)-3/16*e^4*Chi(3*(a+b*arcsinh(c+d*x))/b)*cosh(3*a/b)/(b*d)+1/16*e^4*Chi(5*(a+b*arcsinh(c+d*x))/b)*cosh(5*a/b)/(b*d)-1/8*e^4*Shi((a+b*arcsinh(c+d*x))/b)*sinh(a/b)/(b*d)+3/16*e^4*Shi(3*(a+b*arcsinh(c+d*x))/b)*sinh(3*a/b)/(b*d)-1/16*e^4*Shi(5*(a+b*arcsinh(c+d*x))/b)*sinh(5*a/b)/(b*d)],
[(c*e+d*e*x)^3/(a+b*arcsinh(c+d*x)),x,11,-1/4*e^3*cosh(2*a/b)*Shi(2*(a+b*arcsinh(c+d*x))/b)/(b*d)+1/8*e^3*cosh(4*a/b)*Shi(4*(a+b*arcsinh(c+d*x))/b)/(b*d)+1/4*e^3*Chi(2*(a+b*arcsinh(c+d*x))/b)*sinh(2*a/b)/(b*d)-1/8*e^3*Chi(4*(a+b*arcsinh(c+d*x))/b)*sinh(4*a/b)/(b*d)],
[(c*e+d*e*x)^2/(a+b*arcsinh(c+d*x)),x,11,-1/4*e^2*Chi((a+b*arcsinh(c+d*x))/b)*cosh(a/b)/(b*d)+1/4*e^2*Chi(3*(a+b*arcsinh(c+d*x))/b)*cosh(3*a/b)/(b*d)+1/4*e^2*Shi((a+b*arcsinh(c+d*x))/b)*sinh(a/b)/(b*d)-1/4*e^2*Shi(3*(a+b*arcsinh(c+d*x))/b)*sinh(3*a/b)/(b*d)],
[(c*e+d*e*x)/(a+b*arcsinh(c+d*x)),x,8,1/2*e*cosh(2*a/b)*Shi(2*(a+b*arcsinh(c+d*x))/b)/(b*d)-1/2*e*Chi(2*(a+b*arcsinh(c+d*x))/b)*sinh(2*a/b)/(b*d)],
[1/(a+b*arcsinh(c+d*x)),x,5,Chi((a+b*arcsinh(c+d*x))/b)*cosh(a/b)/(b*d)-Shi((a+b*arcsinh(c+d*x))/b)*sinh(a/b)/(b*d)],
[1/((c*e+d*e*x)*(a+b*arcsinh(c+d*x))),x,2,Unintegrable(1/((c+d*x)*(a+b*arcsinh(c+d*x))),x)/e],
[(c*e+d*e*x)^4/(a+b*arcsinh(c+d*x))^2,x,13,1/8*e^4*cosh(a/b)*Shi((a+b*arcsinh(c+d*x))/b)/(b^2*d)-9/16*e^4*cosh(3*a/b)*Shi(3*(a+b*arcsinh(c+d*x))/b)/(b^2*d)+5/16*e^4*cosh(5*a/b)*Shi(5*(a+b*arcsinh(c+d*x))/b)/(b^2*d)-1/8*e^4*Chi((a+b*arcsinh(c+d*x))/b)*sinh(a/b)/(b^2*d)+9/16*e^4*Chi(3*(a+b*arcsinh(c+d*x))/b)*sinh(3*a/b)/(b^2*d)-5/16*e^4*Chi(5*(a+b*arcsinh(c+d*x))/b)*sinh(5*a/b)/(b^2*d)-e^4*(c+d*x)^4*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)^3/(a+b*arcsinh(c+d*x))^2,x,10,-1/2*e^3*Chi(2*(a+b*arcsinh(c+d*x))/b)*cosh(2*a/b)/(b^2*d)+1/2*e^3*Chi(4*(a+b*arcsinh(c+d*x))/b)*cosh(4*a/b)/(b^2*d)+1/2*e^3*Shi(2*(a+b*arcsinh(c+d*x))/b)*sinh(2*a/b)/(b^2*d)-1/2*e^3*Shi(4*(a+b*arcsinh(c+d*x))/b)*sinh(4*a/b)/(b^2*d)-e^3*(c+d*x)^3*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)^2/(a+b*arcsinh(c+d*x))^2,x,10,-1/4*e^2*cosh(a/b)*Shi((a+b*arcsinh(c+d*x))/b)/(b^2*d)+3/4*e^2*cosh(3*a/b)*Shi(3*(a+b*arcsinh(c+d*x))/b)/(b^2*d)+1/4*e^2*Chi((a+b*arcsinh(c+d*x))/b)*sinh(a/b)/(b^2*d)-3/4*e^2*Chi(3*(a+b*arcsinh(c+d*x))/b)*sinh(3*a/b)/(b^2*d)-e^2*(c+d*x)^2*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)/(a+b*arcsinh(c+d*x))^2,x,6,e*Chi(2*(a+b*arcsinh(c+d*x))/b)*cosh(2*a/b)/(b^2*d)-e*Shi(2*(a+b*arcsinh(c+d*x))/b)*sinh(2*a/b)/(b^2*d)-e*(c+d*x)*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x)))],
[1/(a+b*arcsinh(c+d*x))^2,x,6,cosh(a/b)*Shi((a+b*arcsinh(c+d*x))/b)/(b^2*d)-Chi((a+b*arcsinh(c+d*x))/b)*sinh(a/b)/(b^2*d)-sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x)))],
[1/((c*e+d*e*x)*(a+b*arcsinh(c+d*x))^2),x,2,Unintegrable(1/((c+d*x)*(a+b*arcsinh(c+d*x))^2),x)/e],
[(c*e+d*e*x)^4/(a+b*arcsinh(c+d*x))^3,x,26,-2*e^4*(c+d*x)^3/(b^2*d*(a+b*arcsinh(c+d*x)))-5/2*e^4*(c+d*x)^5/(b^2*d*(a+b*arcsinh(c+d*x)))+1/16*e^4*Chi((a+b*arcsinh(c+d*x))/b)*cosh(a/b)/(b^3*d)-27/32*e^4*Chi(3*(a+b*arcsinh(c+d*x))/b)*cosh(3*a/b)/(b^3*d)+25/32*e^4*Chi(5*(a+b*arcsinh(c+d*x))/b)*cosh(5*a/b)/(b^3*d)-1/16*e^4*Shi((a+b*arcsinh(c+d*x))/b)*sinh(a/b)/(b^3*d)+27/32*e^4*Shi(3*(a+b*arcsinh(c+d*x))/b)*sinh(3*a/b)/(b^3*d)-25/32*e^4*Shi(5*(a+b*arcsinh(c+d*x))/b)*sinh(5*a/b)/(b^3*d)-1/2*e^4*(c+d*x)^4*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^2)],
[(c*e+d*e*x)^3/(a+b*arcsinh(c+d*x))^3,x,20,-3/2*e^3*(c+d*x)^2/(b^2*d*(a+b*arcsinh(c+d*x)))-2*e^3*(c+d*x)^4/(b^2*d*(a+b*arcsinh(c+d*x)))-1/2*e^3*cosh(2*a/b)*Shi(2*(a+b*arcsinh(c+d*x))/b)/(b^3*d)+e^3*cosh(4*a/b)*Shi(4*(a+b*arcsinh(c+d*x))/b)/(b^3*d)+1/2*e^3*Chi(2*(a+b*arcsinh(c+d*x))/b)*sinh(2*a/b)/(b^3*d)-e^3*Chi(4*(a+b*arcsinh(c+d*x))/b)*sinh(4*a/b)/(b^3*d)-1/2*e^3*(c+d*x)^3*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^2)],
[(c*e+d*e*x)^2/(a+b*arcsinh(c+d*x))^3,x,18,-e^2*(c+d*x)/(b^2*d*(a+b*arcsinh(c+d*x)))-3/2*e^2*(c+d*x)^3/(b^2*d*(a+b*arcsinh(c+d*x)))-1/8*e^2*Chi((a+b*arcsinh(c+d*x))/b)*cosh(a/b)/(b^3*d)+9/8*e^2*Chi(3*(a+b*arcsinh(c+d*x))/b)*cosh(3*a/b)/(b^3*d)+1/8*e^2*Shi((a+b*arcsinh(c+d*x))/b)*sinh(a/b)/(b^3*d)-9/8*e^2*Shi(3*(a+b*arcsinh(c+d*x))/b)*sinh(3*a/b)/(b^3*d)-1/2*e^2*(c+d*x)^2*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^2)],
[(c*e+d*e*x)/(a+b*arcsinh(c+d*x))^3,x,11,-1/2*e/(b^2*d*(a+b*arcsinh(c+d*x)))-e*(c+d*x)^2/(b^2*d*(a+b*arcsinh(c+d*x)))+e*cosh(2*a/b)*Shi(2*(a+b*arcsinh(c+d*x))/b)/(b^3*d)-e*Chi(2*(a+b*arcsinh(c+d*x))/b)*sinh(2*a/b)/(b^3*d)-1/2*e*(c+d*x)*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^2)],
[1/(a+b*arcsinh(c+d*x))^3,x,7,1/2*(-c-d*x)/(b^2*d*(a+b*arcsinh(c+d*x)))+1/2*Chi((a+b*arcsinh(c+d*x))/b)*cosh(a/b)/(b^3*d)-1/2*Shi((a+b*arcsinh(c+d*x))/b)*sinh(a/b)/(b^3*d)-1/2*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^2)],
[1/((c*e+d*e*x)*(a+b*arcsinh(c+d*x))^3),x,2,Unintegrable(1/((c+d*x)*(a+b*arcsinh(c+d*x))^3),x)/e],
[(c*e+d*e*x)^4/(a+b*arcsinh(c+d*x))^4,x,24,-2/3*e^4*(c+d*x)^3/(b^2*d*(a+b*arcsinh(c+d*x))^2)-5/6*e^4*(c+d*x)^5/(b^2*d*(a+b*arcsinh(c+d*x))^2)+1/48*e^4*cosh(a/b)*Shi((a+b*arcsinh(c+d*x))/b)/(b^4*d)-27/32*e^4*cosh(3*a/b)*Shi(3*(a+b*arcsinh(c+d*x))/b)/(b^4*d)+125/96*e^4*cosh(5*a/b)*Shi(5*(a+b*arcsinh(c+d*x))/b)/(b^4*d)-1/48*e^4*Chi((a+b*arcsinh(c+d*x))/b)*sinh(a/b)/(b^4*d)+27/32*e^4*Chi(3*(a+b*arcsinh(c+d*x))/b)*sinh(3*a/b)/(b^4*d)-125/96*e^4*Chi(5*(a+b*arcsinh(c+d*x))/b)*sinh(5*a/b)/(b^4*d)-1/3*e^4*(c+d*x)^4*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^3)-2*e^4*(c+d*x)^2*sqrt(1+(c+d*x)^2)/(b^3*d*(a+b*arcsinh(c+d*x)))-25/6*e^4*(c+d*x)^4*sqrt(1+(c+d*x)^2)/(b^3*d*(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)^3/(a+b*arcsinh(c+d*x))^4,x,17,-1/2*e^3*(c+d*x)^2/(b^2*d*(a+b*arcsinh(c+d*x))^2)-2/3*e^3*(c+d*x)^4/(b^2*d*(a+b*arcsinh(c+d*x))^2)-1/3*e^3*Chi(2*(a+b*arcsinh(c+d*x))/b)*cosh(2*a/b)/(b^4*d)+4/3*e^3*Chi(4*(a+b*arcsinh(c+d*x))/b)*cosh(4*a/b)/(b^4*d)+1/3*e^3*Shi(2*(a+b*arcsinh(c+d*x))/b)*sinh(2*a/b)/(b^4*d)-4/3*e^3*Shi(4*(a+b*arcsinh(c+d*x))/b)*sinh(4*a/b)/(b^4*d)-1/3*e^3*(c+d*x)^3*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^3)-e^3*(c+d*x)*sqrt(1+(c+d*x)^2)/(b^3*d*(a+b*arcsinh(c+d*x)))-8/3*e^3*(c+d*x)^3*sqrt(1+(c+d*x)^2)/(b^3*d*(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)^2/(a+b*arcsinh(c+d*x))^4,x,18,-1/3*e^2*(c+d*x)/(b^2*d*(a+b*arcsinh(c+d*x))^2)-1/2*e^2*(c+d*x)^3/(b^2*d*(a+b*arcsinh(c+d*x))^2)-1/24*e^2*cosh(a/b)*Shi((a+b*arcsinh(c+d*x))/b)/(b^4*d)+9/8*e^2*cosh(3*a/b)*Shi(3*(a+b*arcsinh(c+d*x))/b)/(b^4*d)+1/24*e^2*Chi((a+b*arcsinh(c+d*x))/b)*sinh(a/b)/(b^4*d)-9/8*e^2*Chi(3*(a+b*arcsinh(c+d*x))/b)*sinh(3*a/b)/(b^4*d)-1/3*e^2*(c+d*x)^2*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^3)-1/3*e^2*sqrt(1+(c+d*x)^2)/(b^3*d*(a+b*arcsinh(c+d*x)))-3/2*e^2*(c+d*x)^2*sqrt(1+(c+d*x)^2)/(b^3*d*(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)/(a+b*arcsinh(c+d*x))^4,x,9,-1/6*e/(b^2*d*(a+b*arcsinh(c+d*x))^2)-1/3*e*(c+d*x)^2/(b^2*d*(a+b*arcsinh(c+d*x))^2)+2/3*e*Chi(2*(a+b*arcsinh(c+d*x))/b)*cosh(2*a/b)/(b^4*d)-2/3*e*Shi(2*(a+b*arcsinh(c+d*x))/b)*sinh(2*a/b)/(b^4*d)-1/3*e*(c+d*x)*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^3)-2/3*e*(c+d*x)*sqrt(1+(c+d*x)^2)/(b^3*d*(a+b*arcsinh(c+d*x)))],
[1/(a+b*arcsinh(c+d*x))^4,x,8,1/6*(-c-d*x)/(b^2*d*(a+b*arcsinh(c+d*x))^2)+1/6*cosh(a/b)*Shi((a+b*arcsinh(c+d*x))/b)/(b^4*d)-1/6*Chi((a+b*arcsinh(c+d*x))/b)*sinh(a/b)/(b^4*d)-1/3*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^3)-1/6*sqrt(1+(c+d*x)^2)/(b^3*d*(a+b*arcsinh(c+d*x)))],
[1/((c*e+d*e*x)*(a+b*arcsinh(c+d*x))^4),x,2,Unintegrable(1/((c+d*x)*(a+b*arcsinh(c+d*x))^4),x)/e],

# Integrands of the form (c e+d e x)^m (a+b ArcSinh[c+d x])^(n/2)

# n>0
[(c*e+d*e*x)^4*sqrt(a+b*arcsinh(c+d*x)),x,21,1/320*e^4*exp(5*a/b)*erf(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/5*Pi)/d-1/320*e^4*erfi(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/5*Pi)/(d*exp(5*a/b))-1/64*e^4*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/3*Pi)/d+1/64*e^4*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/3*Pi)/(d*exp(3*a/b))+1/32*e^4*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/d-1/32*e^4*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/(d*exp(a/b))+1/5*e^4*(c+d*x)^5*sqrt(a+b*arcsinh(c+d*x))/d],
[(c*e+d*e*x)^3*sqrt(a+b*arcsinh(c+d*x)),x,16,1/32*e^3*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/d+1/32*e^3*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/(d*exp(2*a/b))-1/256*e^3*exp(4*a/b)*erf(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/d-1/256*e^3*erfi(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/(d*exp(4*a/b))-3/32*e^3*sqrt(a+b*arcsinh(c+d*x))/d+1/4*e^3*(c+d*x)^4*sqrt(a+b*arcsinh(c+d*x))/d],
[(c*e+d*e*x)^2*sqrt(a+b*arcsinh(c+d*x)),x,16,1/48*e^2*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/3*Pi)/d-1/48*e^2*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/3*Pi)/(d*exp(3*a/b))-1/16*e^2*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/d+1/16*e^2*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/(d*exp(a/b))+1/3*e^2*(c+d*x)^3*sqrt(a+b*arcsinh(c+d*x))/d],
[(c*e+d*e*x)*sqrt(a+b*arcsinh(c+d*x)),x,11,-1/16*e*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/d-1/16*e*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/(d*exp(2*a/b))+1/4*e*sqrt(a+b*arcsinh(c+d*x))/d+1/2*e*(c+d*x)^2*sqrt(a+b*arcsinh(c+d*x))/d],
[sqrt(a+b*arcsinh(c+d*x)),x,8,1/4*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/d-1/4*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/(d*exp(a/b))+(c+d*x)*sqrt(a+b*arcsinh(c+d*x))/d],
[sqrt(a+b*arcsinh(c+d*x))/(c*e+d*e*x),x,2,Unintegrable(sqrt(a+b*arcsinh(c+d*x))/(c+d*x),x)/e],
[(c*e+d*e*x)^4*(a+b*arcsinh(c+d*x))^(3/2),x,43,1/5*e^4*(c+d*x)^5*(a+b*arcsinh(c+d*x))^(3/2)/d+3/3200*b^(3/2)*e^4*exp(5*a/b)*erf(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/5*Pi)/d+3/3200*b^(3/2)*e^4*erfi(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/5*Pi)/(d*exp(5*a/b))-1/200*b^(3/2)*e^4*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/d-1/200*b^(3/2)*e^4*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/(d*exp(3*a/b))+3/64*b^(3/2)*e^4*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d+3/64*b^(3/2)*e^4*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b))-3/3200*b^(3/2)*e^4*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/d-3/3200*b^(3/2)*e^4*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(d*exp(3*a/b))-4/25*b*e^4*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d+2/25*b*e^4*(c+d*x)^2*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d-3/50*b*e^4*(c+d*x)^4*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d],
[(c*e+d*e*x)^3*(a+b*arcsinh(c+d*x))^(3/2),x,27,-3/32*e^3*(a+b*arcsinh(c+d*x))^(3/2)/d+1/4*e^3*(c+d*x)^4*(a+b*arcsinh(c+d*x))^(3/2)/d+3/128*b^(3/2)*e^3*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d-3/128*b^(3/2)*e^3*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d*exp(2*a/b))-3/2048*b^(3/2)*e^3*exp(4*a/b)*erf(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d+3/2048*b^(3/2)*e^3*erfi(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(4*a/b))+9/64*b*e^3*(c+d*x)*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d-3/32*b*e^3*(c+d*x)^3*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d],
[(c*e+d*e*x)^2*(a+b*arcsinh(c+d*x))^(3/2),x,24,1/3*e^2*(c+d*x)^3*(a+b*arcsinh(c+d*x))^(3/2)/d+1/96*b^(3/2)*e^2*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/d+1/96*b^(3/2)*e^2*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/(d*exp(3*a/b))-3/32*b^(3/2)*e^2*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d-3/32*b^(3/2)*e^2*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b))+1/3*b*e^2*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d-1/6*b*e^2*(c+d*x)^2*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d],
[(c*e+d*e*x)*(a+b*arcsinh(c+d*x))^(3/2),x,13,1/4*e*(a+b*arcsinh(c+d*x))^(3/2)/d+1/2*e*(c+d*x)^2*(a+b*arcsinh(c+d*x))^(3/2)/d-3/64*b^(3/2)*e*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d+3/64*b^(3/2)*e*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d*exp(2*a/b))-3/8*b*e*(c+d*x)*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d],
[(a+b*arcsinh(c+d*x))^(3/2),x,9,(c+d*x)*(a+b*arcsinh(c+d*x))^(3/2)/d+3/8*b^(3/2)*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d+3/8*b^(3/2)*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b))-3/2*b*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d],
[(a+b*arcsinh(c+d*x))^(3/2)/(c*e+d*e*x),x,2,Unintegrable((a+b*arcsinh(c+d*x))^(3/2)/(c+d*x),x)/e],
[(c*e+d*e*x)^4*(a+b*arcsinh(c+d*x))^(5/2),x,46,1/5*e^4*(c+d*x)^5*(a+b*arcsinh(c+d*x))^(5/2)/d+3/6400*b^(5/2)*e^4*exp(5*a/b)*erf(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/5*Pi)/d-3/6400*b^(5/2)*e^4*erfi(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/5*Pi)/(d*exp(5*a/b))-1/240*b^(5/2)*e^4*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/d+1/240*b^(5/2)*e^4*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/(d*exp(3*a/b))+15/128*b^(5/2)*e^4*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d-15/128*b^(5/2)*e^4*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b))-1/1280*b^(5/2)*e^4*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/d+1/1280*b^(5/2)*e^4*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(d*exp(3*a/b))-4/15*b*e^4*(a+b*arcsinh(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)/d+2/15*b*e^4*(c+d*x)^2*(a+b*arcsinh(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)/d-1/10*b*e^4*(c+d*x)^4*(a+b*arcsinh(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)/d+2/5*b^2*e^4*(c+d*x)*sqrt(a+b*arcsinh(c+d*x))/d-1/15*b^2*e^4*(c+d*x)^3*sqrt(a+b*arcsinh(c+d*x))/d+3/100*b^2*e^4*(c+d*x)^5*sqrt(a+b*arcsinh(c+d*x))/d],
[(c*e+d*e*x)^3*(a+b*arcsinh(c+d*x))^(5/2),x,29,-3/32*e^3*(a+b*arcsinh(c+d*x))^(5/2)/d+1/4*e^3*(c+d*x)^4*(a+b*arcsinh(c+d*x))^(5/2)/d+15/512*b^(5/2)*e^3*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d+15/512*b^(5/2)*e^3*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d*exp(2*a/b))-15/16384*b^(5/2)*e^3*exp(4*a/b)*erf(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d-15/16384*b^(5/2)*e^3*erfi(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(4*a/b))+15/64*b*e^3*(c+d*x)*(a+b*arcsinh(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)/d-5/32*b*e^3*(c+d*x)^3*(a+b*arcsinh(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)/d-225/2048*b^2*e^3*sqrt(a+b*arcsinh(c+d*x))/d-45/256*b^2*e^3*(c+d*x)^2*sqrt(a+b*arcsinh(c+d*x))/d+15/256*b^2*e^3*(c+d*x)^4*sqrt(a+b*arcsinh(c+d*x))/d],
[(c*e+d*e*x)^2*(a+b*arcsinh(c+d*x))^(5/2),x,26,1/3*e^2*(c+d*x)^3*(a+b*arcsinh(c+d*x))^(5/2)/d+5/576*b^(5/2)*e^2*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/d-5/576*b^(5/2)*e^2*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/(d*exp(3*a/b))-15/64*b^(5/2)*e^2*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d+15/64*b^(5/2)*e^2*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b))+5/9*b*e^2*(a+b*arcsinh(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)/d-5/18*b*e^2*(c+d*x)^2*(a+b*arcsinh(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)/d-5/6*b^2*e^2*(c+d*x)*sqrt(a+b*arcsinh(c+d*x))/d+5/36*b^2*e^2*(c+d*x)^3*sqrt(a+b*arcsinh(c+d*x))/d],
[(c*e+d*e*x)*(a+b*arcsinh(c+d*x))^(5/2),x,14,1/4*e*(a+b*arcsinh(c+d*x))^(5/2)/d+1/2*e*(c+d*x)^2*(a+b*arcsinh(c+d*x))^(5/2)/d-15/256*b^(5/2)*e*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d-15/256*b^(5/2)*e*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d*exp(2*a/b))-5/8*b*e*(c+d*x)*(a+b*arcsinh(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)/d+15/64*b^2*e*sqrt(a+b*arcsinh(c+d*x))/d+15/32*b^2*e*(c+d*x)^2*sqrt(a+b*arcsinh(c+d*x))/d],
[(a+b*arcsinh(c+d*x))^(5/2),x,10,(c+d*x)*(a+b*arcsinh(c+d*x))^(5/2)/d+15/16*b^(5/2)*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d-15/16*b^(5/2)*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b))-5/2*b*(a+b*arcsinh(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)/d+15/4*b^2*(c+d*x)*sqrt(a+b*arcsinh(c+d*x))/d],
[(a+b*arcsinh(c+d*x))^(5/2)/(c*e+d*e*x),x,2,Unintegrable((a+b*arcsinh(c+d*x))^(5/2)/(c+d*x),x)/e],
[(c*e+d*e*x)^4*(a+b*arcsinh(c+d*x))^(7/2),x,77,14/15*b^2*e^4*(c+d*x)*(a+b*arcsinh(c+d*x))^(3/2)/d-7/45*b^2*e^4*(c+d*x)^3*(a+b*arcsinh(c+d*x))^(3/2)/d+7/100*b^2*e^4*(c+d*x)^5*(a+b*arcsinh(c+d*x))^(3/2)/d+1/5*e^4*(c+d*x)^5*(a+b*arcsinh(c+d*x))^(7/2)/d+21/64000*b^(7/2)*e^4*exp(5*a/b)*erf(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/5*Pi)/d+21/64000*b^(7/2)*e^4*erfi(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/5*Pi)/(d*exp(5*a/b))-119/18000*b^(7/2)*e^4*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/d-119/18000*b^(7/2)*e^4*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/(d*exp(3*a/b))+105/256*b^(7/2)*e^4*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d+105/256*b^(7/2)*e^4*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b))-21/64000*b^(7/2)*e^4*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/d-21/64000*b^(7/2)*e^4*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(d*exp(3*a/b))-28/75*b*e^4*(a+b*arcsinh(c+d*x))^(5/2)*sqrt(1+(c+d*x)^2)/d+14/75*b*e^4*(c+d*x)^2*(a+b*arcsinh(c+d*x))^(5/2)*sqrt(1+(c+d*x)^2)/d-7/50*b*e^4*(c+d*x)^4*(a+b*arcsinh(c+d*x))^(5/2)*sqrt(1+(c+d*x)^2)/d-1813/1125*b^3*e^4*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d+119/1125*b^3*e^4*(c+d*x)^2*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d-21/1000*b^3*e^4*(c+d*x)^4*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d],
[(c*e+d*e*x)^3*(a+b*arcsinh(c+d*x))^(7/2),x,42,-525/2048*b^2*e^3*(a+b*arcsinh(c+d*x))^(3/2)/d-105/256*b^2*e^3*(c+d*x)^2*(a+b*arcsinh(c+d*x))^(3/2)/d+35/256*b^2*e^3*(c+d*x)^4*(a+b*arcsinh(c+d*x))^(3/2)/d-3/32*e^3*(a+b*arcsinh(c+d*x))^(7/2)/d+1/4*e^3*(c+d*x)^4*(a+b*arcsinh(c+d*x))^(7/2)/d+105/2048*b^(7/2)*e^3*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d-105/2048*b^(7/2)*e^3*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d*exp(2*a/b))-105/131072*b^(7/2)*e^3*exp(4*a/b)*erf(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d+105/131072*b^(7/2)*e^3*erfi(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(4*a/b))+21/64*b*e^3*(c+d*x)*(a+b*arcsinh(c+d*x))^(5/2)*sqrt(1+(c+d*x)^2)/d-7/32*b*e^3*(c+d*x)^3*(a+b*arcsinh(c+d*x))^(5/2)*sqrt(1+(c+d*x)^2)/d+1575/4096*b^3*e^3*(c+d*x)*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d-105/2048*b^3*e^3*(c+d*x)^3*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d],
[(c*e+d*e*x)^2*(a+b*arcsinh(c+d*x))^(7/2),x,35,-35/18*b^2*e^2*(c+d*x)*(a+b*arcsinh(c+d*x))^(3/2)/d+35/108*b^2*e^2*(c+d*x)^3*(a+b*arcsinh(c+d*x))^(3/2)/d+1/3*e^2*(c+d*x)^3*(a+b*arcsinh(c+d*x))^(7/2)/d+35/3456*b^(7/2)*e^2*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/d+35/3456*b^(7/2)*e^2*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/(d*exp(3*a/b))-105/128*b^(7/2)*e^2*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d-105/128*b^(7/2)*e^2*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b))+7/9*b*e^2*(a+b*arcsinh(c+d*x))^(5/2)*sqrt(1+(c+d*x)^2)/d-7/18*b*e^2*(c+d*x)^2*(a+b*arcsinh(c+d*x))^(5/2)*sqrt(1+(c+d*x)^2)/d+175/54*b^3*e^2*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d-35/216*b^3*e^2*(c+d*x)^2*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d],
[(c*e+d*e*x)*(a+b*arcsinh(c+d*x))^(7/2),x,16,35/64*b^2*e*(a+b*arcsinh(c+d*x))^(3/2)/d+35/32*b^2*e*(c+d*x)^2*(a+b*arcsinh(c+d*x))^(3/2)/d+1/4*e*(a+b*arcsinh(c+d*x))^(7/2)/d+1/2*e*(c+d*x)^2*(a+b*arcsinh(c+d*x))^(7/2)/d-105/1024*b^(7/2)*e*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d+105/1024*b^(7/2)*e*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d*exp(2*a/b))-7/8*b*e*(c+d*x)*(a+b*arcsinh(c+d*x))^(5/2)*sqrt(1+(c+d*x)^2)/d-105/128*b^3*e*(c+d*x)*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d],
[(a+b*arcsinh(c+d*x))^(7/2),x,11,35/4*b^2*(c+d*x)*(a+b*arcsinh(c+d*x))^(3/2)/d+(c+d*x)*(a+b*arcsinh(c+d*x))^(7/2)/d+105/32*b^(7/2)*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/d+105/32*b^(7/2)*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b))-7/2*b*(a+b*arcsinh(c+d*x))^(5/2)*sqrt(1+(c+d*x)^2)/d-105/8*b^3*sqrt(1+(c+d*x)^2)*sqrt(a+b*arcsinh(c+d*x))/d],
[(a+b*arcsinh(c+d*x))^(7/2)/(c*e+d*e*x),x,2,Unintegrable((a+b*arcsinh(c+d*x))^(7/2)/(c+d*x),x)/e],

# n<0
[(c*e+d*e*x)^4/sqrt(a+b*arcsinh(c+d*x)),x,20,1/32*e^4*exp(5*a/b)*erf(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/5*Pi)/(d*sqrt(b))+1/32*e^4*erfi(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/5*Pi)/(d*exp(5*a/b)*sqrt(b))+1/16*e^4*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*sqrt(b))+1/16*e^4*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b)*sqrt(b))-1/32*e^4*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(d*sqrt(b))-1/32*e^4*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(d*exp(3*a/b)*sqrt(b))],
[(c*e+d*e*x)^3/sqrt(a+b*arcsinh(c+d*x)),x,15,1/8*e^3*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d*sqrt(b))-1/8*e^3*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d*exp(2*a/b)*sqrt(b))-1/32*e^3*exp(4*a/b)*erf(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*sqrt(b))+1/32*e^3*erfi(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(4*a/b)*sqrt(b))],
[(c*e+d*e*x)^2/sqrt(a+b*arcsinh(c+d*x)),x,15,1/8*e^2*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/(d*sqrt(b))+1/8*e^2*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/3*Pi)/(d*exp(3*a/b)*sqrt(b))-1/8*e^2*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*sqrt(b))-1/8*e^2*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b)*sqrt(b))],
[(c*e+d*e*x)/sqrt(a+b*arcsinh(c+d*x)),x,10,-1/4*e*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d*sqrt(b))+1/4*e*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d*exp(2*a/b)*sqrt(b))],
[1/sqrt(a+b*arcsinh(c+d*x)),x,7,1/2*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*sqrt(b))+1/2*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(d*exp(a/b)*sqrt(b))],
[1/((c*e+d*e*x)*sqrt(a+b*arcsinh(c+d*x))),x,2,Unintegrable(1/((c+d*x)*sqrt(a+b*arcsinh(c+d*x))),x)/e],
[(c*e+d*e*x)^4/(a+b*arcsinh(c+d*x))^(3/2),x,19,-1/8*e^4*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*d)+1/8*e^4*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*d*exp(a/b))+3/16*e^4*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(b^(3/2)*d)-3/16*e^4*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(b^(3/2)*d*exp(3*a/b))-1/16*e^4*exp(5*a/b)*erf(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(5*Pi)/(b^(3/2)*d)+1/16*e^4*erfi(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(5*Pi)/(b^(3/2)*d*exp(5*a/b))-2*e^4*(c+d*x)^4*sqrt(1+(c+d*x)^2)/(b*d*sqrt(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)^3/(a+b*arcsinh(c+d*x))^(3/2),x,14,-1/2*e^3*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(b^(3/2)*d)-1/2*e^3*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(b^(3/2)*d*exp(2*a/b))+1/4*e^3*exp(4*a/b)*erf(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*d)+1/4*e^3*erfi(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*d*exp(4*a/b))-2*e^3*(c+d*x)^3*sqrt(1+(c+d*x)^2)/(b*d*sqrt(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)^2/(a+b*arcsinh(c+d*x))^(3/2),x,14,1/4*e^2*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*d)-1/4*e^2*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*d*exp(a/b))-1/4*e^2*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(b^(3/2)*d)+1/4*e^2*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(b^(3/2)*d*exp(3*a/b))-2*e^2*(c+d*x)^2*sqrt(1+(c+d*x)^2)/(b*d*sqrt(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)/(a+b*arcsinh(c+d*x))^(3/2),x,8,e*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(b^(3/2)*d)+e*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(b^(3/2)*d*exp(2*a/b))-2*e*(c+d*x)*sqrt(1+(c+d*x)^2)/(b*d*sqrt(a+b*arcsinh(c+d*x)))],
[1/(a+b*arcsinh(c+d*x))^(3/2),x,8,-exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*d)+erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*d*exp(a/b))-2*sqrt(1+(c+d*x)^2)/(b*d*sqrt(a+b*arcsinh(c+d*x)))],
[1/((c*e+d*e*x)*(a+b*arcsinh(c+d*x))^(3/2)),x,2,Unintegrable(1/((c+d*x)*(a+b*arcsinh(c+d*x))^(3/2)),x)/e],
[(c*e+d*e*x)^4/(a+b*arcsinh(c+d*x))^(5/2),x,36,1/12*e^4*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*d)+1/12*e^4*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*d*exp(a/b))-3/8*e^4*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(b^(5/2)*d)-3/8*e^4*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(b^(5/2)*d*exp(3*a/b))+5/24*e^4*exp(5*a/b)*erf(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(5*Pi)/(b^(5/2)*d)+5/24*e^4*erfi(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(5*Pi)/(b^(5/2)*d*exp(5*a/b))-2/3*e^4*(c+d*x)^4*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^(3/2))-16/3*e^4*(c+d*x)^3/(b^2*d*sqrt(a+b*arcsinh(c+d*x)))-20/3*e^4*(c+d*x)^5/(b^2*d*sqrt(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)^3/(a+b*arcsinh(c+d*x))^(5/2),x,26,-2/3*e^3*exp(4*a/b)*erf(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*d)+2/3*e^3*erfi(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*d*exp(4*a/b))+1/3*e^3*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(5/2)*d)-1/3*e^3*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(5/2)*d*exp(2*a/b))-2/3*e^3*(c+d*x)^3*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^(3/2))-4*e^3*(c+d*x)^2/(b^2*d*sqrt(a+b*arcsinh(c+d*x)))-16/3*e^3*(c+d*x)^4/(b^2*d*sqrt(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)^2/(a+b*arcsinh(c+d*x))^(5/2),x,24,-1/6*e^2*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*d)-1/6*e^2*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*d*exp(a/b))+1/2*e^2*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(b^(5/2)*d)+1/2*e^2*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(b^(5/2)*d*exp(3*a/b))-2/3*e^2*(c+d*x)^2*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^(3/2))-8/3*e^2*(c+d*x)/(b^2*d*sqrt(a+b*arcsinh(c+d*x)))-4*e^2*(c+d*x)^3/(b^2*d*sqrt(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)/(a+b*arcsinh(c+d*x))^(5/2),x,13,-2/3*e*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(5/2)*d)+2/3*e*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(5/2)*d*exp(2*a/b))-2/3*e*(c+d*x)*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^(3/2))-4/3*e/(b^2*d*sqrt(a+b*arcsinh(c+d*x)))-8/3*e*(c+d*x)^2/(b^2*d*sqrt(a+b*arcsinh(c+d*x)))],
[1/(a+b*arcsinh(c+d*x))^(5/2),x,9,2/3*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*d)+2/3*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*d*exp(a/b))-2/3*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^(3/2))-4/3*(c+d*x)/(b^2*d*sqrt(a+b*arcsinh(c+d*x)))],
[1/((c*e+d*e*x)*(a+b*arcsinh(c+d*x))^(5/2)),x,2,Unintegrable(1/((c+d*x)*(a+b*arcsinh(c+d*x))^(5/2)),x)/e],
[(c*e+d*e*x)^4/(a+b*arcsinh(c+d*x))^(7/2),x,34,-16/15*e^4*(c+d*x)^3/(b^2*d*(a+b*arcsinh(c+d*x))^(3/2))-4/3*e^4*(c+d*x)^5/(b^2*d*(a+b*arcsinh(c+d*x))^(3/2))-1/30*e^4*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*d)+1/30*e^4*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*d*exp(a/b))+9/20*e^4*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(b^(7/2)*d)-9/20*e^4*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(b^(7/2)*d*exp(3*a/b))-5/12*e^4*exp(5*a/b)*erf(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(5*Pi)/(b^(7/2)*d)+5/12*e^4*erfi(sqrt(5)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(5*Pi)/(b^(7/2)*d*exp(5*a/b))-2/5*e^4*(c+d*x)^4*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^(5/2))-32/5*e^4*(c+d*x)^2*sqrt(1+(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsinh(c+d*x)))-40/3*e^4*(c+d*x)^4*sqrt(1+(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)^3/(a+b*arcsinh(c+d*x))^(7/2),x,23,-4/5*e^3*(c+d*x)^2/(b^2*d*(a+b*arcsinh(c+d*x))^(3/2))-16/15*e^3*(c+d*x)^4/(b^2*d*(a+b*arcsinh(c+d*x))^(3/2))+16/15*e^3*exp(4*a/b)*erf(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*d)+16/15*e^3*erfi(2*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*d*exp(4*a/b))-4/15*e^3*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(7/2)*d)-4/15*e^3*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(7/2)*d*exp(2*a/b))-2/5*e^3*(c+d*x)^3*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^(5/2))-16/5*e^3*(c+d*x)*sqrt(1+(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsinh(c+d*x)))-128/15*e^3*(c+d*x)^3*sqrt(1+(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)^2/(a+b*arcsinh(c+d*x))^(7/2),x,24,-8/15*e^2*(c+d*x)/(b^2*d*(a+b*arcsinh(c+d*x))^(3/2))-4/5*e^2*(c+d*x)^3/(b^2*d*(a+b*arcsinh(c+d*x))^(3/2))+1/15*e^2*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*d)-1/15*e^2*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*d*exp(a/b))-3/5*e^2*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(b^(7/2)*d)+3/5*e^2*erfi(sqrt(3)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(3*Pi)/(b^(7/2)*d*exp(3*a/b))-2/5*e^2*(c+d*x)^2*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^(5/2))-16/15*e^2*sqrt(1+(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsinh(c+d*x)))-24/5*e^2*(c+d*x)^2*sqrt(1+(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsinh(c+d*x)))],
[(c*e+d*e*x)/(a+b*arcsinh(c+d*x))^(7/2),x,11,-4/15*e/(b^2*d*(a+b*arcsinh(c+d*x))^(3/2))-8/15*e*(c+d*x)^2/(b^2*d*(a+b*arcsinh(c+d*x))^(3/2))+8/15*e*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(7/2)*d)+8/15*e*erfi(sqrt(2)*sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(7/2)*d*exp(2*a/b))-2/5*e*(c+d*x)*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^(5/2))-32/15*e*(c+d*x)*sqrt(1+(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsinh(c+d*x)))],
[1/(a+b*arcsinh(c+d*x))^(7/2),x,10,-4/15*(c+d*x)/(b^2*d*(a+b*arcsinh(c+d*x))^(3/2))-4/15*exp(a/b)*erf(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*d)+4/15*erfi(sqrt(a+b*arcsinh(c+d*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*d*exp(a/b))-2/5*sqrt(1+(c+d*x)^2)/(b*d*(a+b*arcsinh(c+d*x))^(5/2))-8/15*sqrt(1+(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsinh(c+d*x)))],
[1/((c*e+d*e*x)*(a+b*arcsinh(c+d*x))^(7/2)),x,2,Unintegrable(1/((c+d*x)*(a+b*arcsinh(c+d*x))^(7/2)),x)/e],

# Integrands of the form (c e+d e x)^(m/2) (a+b ArcSinh[c+d x])^n

# n>0
[(c*e+d*e*x)^(7/2)*(a+b*arcsinh(c+d*x)),x,8,2/9*(e*(c+d*x))^(9/2)*(a+b*arcsinh(c+d*x))/(d*e)+28/405*b*e^2*(e*(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)/d-4/81*b*(e*(c+d*x))^(7/2)*sqrt(1+(c+d*x)^2)/d-28/135*b*e^3*sqrt(e*(c+d*x))*sqrt(1+(c+d*x)^2)/(d*(1+c+d*x))+28/135*b*e^(7/2)*(1+c+d*x)*sqrt(cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))^2)/cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))*EllipticE(sin(2*arctan(sqrt(e*(c+d*x))/sqrt(e))),sqrt(1/2))*sqrt((1+(c+d*x)^2)/(1+c+d*x)^2)/(d*sqrt(1+(c+d*x)^2))-14/135*b*e^(7/2)*(1+c+d*x)*sqrt(cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))^2)/cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))*EllipticF(sin(2*arctan(sqrt(e*(c+d*x))/sqrt(e))),sqrt(1/2))*sqrt((1+(c+d*x)^2)/(1+c+d*x)^2)/(d*sqrt(1+(c+d*x)^2))],
[(c*e+d*e*x)^(5/2)*(a+b*arcsinh(c+d*x)),x,6,2/7*(e*(c+d*x))^(7/2)*(a+b*arcsinh(c+d*x))/(d*e)-4/49*b*(e*(c+d*x))^(5/2)*sqrt(1+(c+d*x)^2)/d+20/147*b*e^2*sqrt(e*(c+d*x))*sqrt(1+(c+d*x)^2)/d-10/147*b*e^(5/2)*(1+c+d*x)*sqrt(cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))^2)/cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))*EllipticF(sin(2*arctan(sqrt(e*(c+d*x))/sqrt(e))),sqrt(1/2))*sqrt((1+(c+d*x)^2)/(1+c+d*x)^2)/(d*sqrt(1+(c+d*x)^2))],
[(c*e+d*e*x)^(3/2)*(a+b*arcsinh(c+d*x)),x,7,2/5*(e*(c+d*x))^(5/2)*(a+b*arcsinh(c+d*x))/(d*e)-4/25*b*(e*(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)/d+12/25*b*e*sqrt(e*(c+d*x))*sqrt(1+(c+d*x)^2)/(d*(1+c+d*x))-12/25*b*e^(3/2)*(1+c+d*x)*sqrt(cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))^2)/cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))*EllipticE(sin(2*arctan(sqrt(e*(c+d*x))/sqrt(e))),sqrt(1/2))*sqrt((1+(c+d*x)^2)/(1+c+d*x)^2)/(d*sqrt(1+(c+d*x)^2))+6/25*b*e^(3/2)*(1+c+d*x)*sqrt(cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))^2)/cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))*EllipticF(sin(2*arctan(sqrt(e*(c+d*x))/sqrt(e))),sqrt(1/2))*sqrt((1+(c+d*x)^2)/(1+c+d*x)^2)/(d*sqrt(1+(c+d*x)^2))],
[(a+b*arcsinh(c+d*x))*sqrt(c*e+d*e*x),x,5,2/3*(e*(c+d*x))^(3/2)*(a+b*arcsinh(c+d*x))/(d*e)-4/9*b*sqrt(e*(c+d*x))*sqrt(1+(c+d*x)^2)/d+2/9*b*(1+c+d*x)*sqrt(cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))^2)/cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))*EllipticF(sin(2*arctan(sqrt(e*(c+d*x))/sqrt(e))),sqrt(1/2))*sqrt(e)*sqrt((1+(c+d*x)^2)/(1+c+d*x)^2)/(d*sqrt(1+(c+d*x)^2))],
[(a+b*arcsinh(c+d*x))/sqrt(c*e+d*e*x),x,6,2*(a+b*arcsinh(c+d*x))*sqrt(e*(c+d*x))/(d*e)-4*b*sqrt(e*(c+d*x))*sqrt(1+(c+d*x)^2)/(d*e*(1+c+d*x))+4*b*(1+c+d*x)*sqrt(cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))^2)/cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))*EllipticE(sin(2*arctan(sqrt(e*(c+d*x))/sqrt(e))),sqrt(1/2))*sqrt((1+(c+d*x)^2)/(1+c+d*x)^2)/(d*sqrt(e)*sqrt(1+(c+d*x)^2))-2*b*(1+c+d*x)*sqrt(cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))^2)/cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))*EllipticF(sin(2*arctan(sqrt(e*(c+d*x))/sqrt(e))),sqrt(1/2))*sqrt((1+(c+d*x)^2)/(1+c+d*x)^2)/(d*sqrt(e)*sqrt(1+(c+d*x)^2))],
[(a+b*arcsinh(c+d*x))/(c*e+d*e*x)^(3/2),x,4,-2*(a+b*arcsinh(c+d*x))/(d*e*sqrt(e*(c+d*x)))+2*b*(1+c+d*x)*sqrt(cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))^2)/cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))*EllipticF(sin(2*arctan(sqrt(e*(c+d*x))/sqrt(e))),sqrt(1/2))*sqrt((1+(c+d*x)^2)/(1+c+d*x)^2)/(d*e^(3/2)*sqrt(1+(c+d*x)^2))],
[(a+b*arcsinh(c+d*x))/(c*e+d*e*x)^(5/2),x,7,-2/3*(a+b*arcsinh(c+d*x))/(d*e*(e*(c+d*x))^(3/2))-4/3*b*sqrt(1+(c+d*x)^2)/(d*e^2*sqrt(e*(c+d*x)))+4/3*b*sqrt(e*(c+d*x))*sqrt(1+(c+d*x)^2)/(d*e^3*(1+c+d*x))-4/3*b*(1+c+d*x)*sqrt(cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))^2)/cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))*EllipticE(sin(2*arctan(sqrt(e*(c+d*x))/sqrt(e))),sqrt(1/2))*sqrt((1+(c+d*x)^2)/(1+c+d*x)^2)/(d*e^(5/2)*sqrt(1+(c+d*x)^2))+2/3*b*(1+c+d*x)*sqrt(cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))^2)/cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))*EllipticF(sin(2*arctan(sqrt(e*(c+d*x))/sqrt(e))),sqrt(1/2))*sqrt((1+(c+d*x)^2)/(1+c+d*x)^2)/(d*e^(5/2)*sqrt(1+(c+d*x)^2))],
[(a+b*arcsinh(c+d*x))/(c*e+d*e*x)^(7/2),x,5,-2/5*(a+b*arcsinh(c+d*x))/(d*e*(e*(c+d*x))^(5/2))-4/15*b*sqrt(1+(c+d*x)^2)/(d*e^2*(e*(c+d*x))^(3/2))-2/15*b*(1+c+d*x)*sqrt(cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))^2)/cos(2*arctan(sqrt(e*(c+d*x))/sqrt(e)))*EllipticF(sin(2*arctan(sqrt(e*(c+d*x))/sqrt(e))),sqrt(1/2))*sqrt((1+(c+d*x)^2)/(1+c+d*x)^2)/(d*e^(7/2)*sqrt(1+(c+d*x)^2))],
[(c*e+d*e*x)^(7/2)*(a+b*arcsinh(c+d*x))^2,x,3,2/9*(e*(c+d*x))^(9/2)*(a+b*arcsinh(c+d*x))^2/(d*e)-8/99*b*(e*(c+d*x))^(11/2)*(a+b*arcsinh(c+d*x))*hypergeom([1/2,11/4],[15/4],-(c+d*x)^2)/(d*e^2)+16/1287*b^2*(e*(c+d*x))^(13/2)*HypergeometricPFQ([1,13/4,13/4],[15/4,17/4],-(c+d*x)^2)/(d*e^3)],
[(c*e+d*e*x)^(5/2)*(a+b*arcsinh(c+d*x))^2,x,3,2/7*(e*(c+d*x))^(7/2)*(a+b*arcsinh(c+d*x))^2/(d*e)-8/63*b*(e*(c+d*x))^(9/2)*(a+b*arcsinh(c+d*x))*hypergeom([1/2,9/4],[13/4],-(c+d*x)^2)/(d*e^2)+16/693*b^2*(e*(c+d*x))^(11/2)*HypergeometricPFQ([1,11/4,11/4],[13/4,15/4],-(c+d*x)^2)/(d*e^3)],
[(c*e+d*e*x)^(3/2)*(a+b*arcsinh(c+d*x))^2,x,3,2/5*(e*(c+d*x))^(5/2)*(a+b*arcsinh(c+d*x))^2/(d*e)-8/35*b*(e*(c+d*x))^(7/2)*(a+b*arcsinh(c+d*x))*hypergeom([1/2,7/4],[11/4],-(c+d*x)^2)/(d*e^2)+16/315*b^2*(e*(c+d*x))^(9/2)*HypergeometricPFQ([1,9/4,9/4],[11/4,13/4],-(c+d*x)^2)/(d*e^3)],
[(a+b*arcsinh(c+d*x))^2*sqrt(c*e+d*e*x),x,3,2/3*(e*(c+d*x))^(3/2)*(a+b*arcsinh(c+d*x))^2/(d*e)-8/15*b*(e*(c+d*x))^(5/2)*(a+b*arcsinh(c+d*x))*hypergeom([1/2,5/4],[9/4],-(c+d*x)^2)/(d*e^2)+16/105*b^2*(e*(c+d*x))^(7/2)*HypergeometricPFQ([1,7/4,7/4],[9/4,11/4],-(c+d*x)^2)/(d*e^3)],
[(a+b*arcsinh(c+d*x))^2/sqrt(c*e+d*e*x),x,3,-8/3*b*(e*(c+d*x))^(3/2)*(a+b*arcsinh(c+d*x))*hypergeom([1/2,3/4],[7/4],-(c+d*x)^2)/(d*e^2)+16/15*b^2*(e*(c+d*x))^(5/2)*HypergeometricPFQ([1,5/4,5/4],[7/4,9/4],-(c+d*x)^2)/(d*e^3)+2*(a+b*arcsinh(c+d*x))^2*sqrt(e*(c+d*x))/(d*e)],
[(a+b*arcsinh(c+d*x))^2/(c*e+d*e*x)^(3/2),x,3,-16/3*b^2*(e*(c+d*x))^(3/2)*HypergeometricPFQ([3/4,3/4,1],[5/4,7/4],-(c+d*x)^2)/(d*e^3)-2*(a+b*arcsinh(c+d*x))^2/(d*e*sqrt(e*(c+d*x)))+8*b*(a+b*arcsinh(c+d*x))*hypergeom([1/4,1/2],[5/4],-(c+d*x)^2)*sqrt(e*(c+d*x))/(d*e^2)],
[(a+b*arcsinh(c+d*x))^2/(c*e+d*e*x)^(5/2),x,3,-2/3*(a+b*arcsinh(c+d*x))^2/(d*e*(e*(c+d*x))^(3/2))-8/3*b*(a+b*arcsinh(c+d*x))*hypergeom([-1/4,1/2],[3/4],-(c+d*x)^2)/(d*e^2*sqrt(e*(c+d*x)))+16/3*b^2*HypergeometricPFQ([1/4,1/4,1],[3/4,5/4],-(c+d*x)^2)*sqrt(e*(c+d*x))/(d*e^3)],
[(a+b*arcsinh(c+d*x))^2/(c*e+d*e*x)^(7/2),x,3,-2/5*(a+b*arcsinh(c+d*x))^2/(d*e*(e*(c+d*x))^(5/2))-8/15*b*(a+b*arcsinh(c+d*x))*hypergeom([-3/4,1/2],[1/4],-(c+d*x)^2)/(d*e^2*(e*(c+d*x))^(3/2))-16/15*b^2*HypergeometricPFQ([-1/4,-1/4,1],[1/4,3/4],-(c+d*x)^2)/(d*e^3*sqrt(e*(c+d*x)))],
[(c*e+d*e*x)^(7/2)*(a+b*arcsinh(c+d*x))^3,x,2,2/9*(e*(c+d*x))^(9/2)*(a+b*arcsinh(c+d*x))^3/(d*e)-2/3*b*Unintegrable((e*(c+d*x))^(9/2)*(a+b*arcsinh(c+d*x))^2/sqrt(1+(c+d*x)^2),x)/e],
[(c*e+d*e*x)^(5/2)*(a+b*arcsinh(c+d*x))^3,x,2,2/7*(e*(c+d*x))^(7/2)*(a+b*arcsinh(c+d*x))^3/(d*e)-6/7*b*Unintegrable((e*(c+d*x))^(7/2)*(a+b*arcsinh(c+d*x))^2/sqrt(1+(c+d*x)^2),x)/e],
[(c*e+d*e*x)^(3/2)*(a+b*arcsinh(c+d*x))^3,x,2,2/5*(e*(c+d*x))^(5/2)*(a+b*arcsinh(c+d*x))^3/(d*e)-6/5*b*Unintegrable((e*(c+d*x))^(5/2)*(a+b*arcsinh(c+d*x))^2/sqrt(1+(c+d*x)^2),x)/e],
[(a+b*arcsinh(c+d*x))^3*sqrt(c*e+d*e*x),x,2,2/3*(e*(c+d*x))^(3/2)*(a+b*arcsinh(c+d*x))^3/(d*e)-2*b*Unintegrable((e*(c+d*x))^(3/2)*(a+b*arcsinh(c+d*x))^2/sqrt(1+(c+d*x)^2),x)/e],
[(a+b*arcsinh(c+d*x))^3/sqrt(c*e+d*e*x),x,2,2*(a+b*arcsinh(c+d*x))^3*sqrt(e*(c+d*x))/(d*e)-6*b*Unintegrable((a+b*arcsinh(c+d*x))^2*sqrt(e*(c+d*x))/sqrt(1+(c+d*x)^2),x)/e],
[(a+b*arcsinh(c+d*x))^3/(c*e+d*e*x)^(3/2),x,2,-2*(a+b*arcsinh(c+d*x))^3/(d*e*sqrt(e*(c+d*x)))+6*b*Unintegrable((a+b*arcsinh(c+d*x))^2/(sqrt(e*(c+d*x))*sqrt(1+(c+d*x)^2)),x)/e],
[(a+b*arcsinh(c+d*x))^3/(c*e+d*e*x)^(5/2),x,2,-2/3*(a+b*arcsinh(c+d*x))^3/(d*e*(e*(c+d*x))^(3/2))+2*b*Unintegrable((a+b*arcsinh(c+d*x))^2/((e*(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)),x)/e],
[(a+b*arcsinh(c+d*x))^3/(c*e+d*e*x)^(7/2),x,2,-2/5*(a+b*arcsinh(c+d*x))^3/(d*e*(e*(c+d*x))^(5/2))+6/5*b*Unintegrable((a+b*arcsinh(c+d*x))^2/((e*(c+d*x))^(5/2)*sqrt(1+(c+d*x)^2)),x)/e],
[(c*e+d*e*x)^(7/2)*(a+b*arcsinh(c+d*x))^4,x,2,2/9*(e*(c+d*x))^(9/2)*(a+b*arcsinh(c+d*x))^4/(d*e)-8/9*b*Unintegrable((e*(c+d*x))^(9/2)*(a+b*arcsinh(c+d*x))^3/sqrt(1+(c+d*x)^2),x)/e],
[(c*e+d*e*x)^(5/2)*(a+b*arcsinh(c+d*x))^4,x,2,2/7*(e*(c+d*x))^(7/2)*(a+b*arcsinh(c+d*x))^4/(d*e)-8/7*b*Unintegrable((e*(c+d*x))^(7/2)*(a+b*arcsinh(c+d*x))^3/sqrt(1+(c+d*x)^2),x)/e],
[(c*e+d*e*x)^(3/2)*(a+b*arcsinh(c+d*x))^4,x,2,2/5*(e*(c+d*x))^(5/2)*(a+b*arcsinh(c+d*x))^4/(d*e)-8/5*b*Unintegrable((e*(c+d*x))^(5/2)*(a+b*arcsinh(c+d*x))^3/sqrt(1+(c+d*x)^2),x)/e],
[(a+b*arcsinh(c+d*x))^4*sqrt(c*e+d*e*x),x,2,2/3*(e*(c+d*x))^(3/2)*(a+b*arcsinh(c+d*x))^4/(d*e)-8/3*b*Unintegrable((e*(c+d*x))^(3/2)*(a+b*arcsinh(c+d*x))^3/sqrt(1+(c+d*x)^2),x)/e],
[(a+b*arcsinh(c+d*x))^4/sqrt(c*e+d*e*x),x,2,2*(a+b*arcsinh(c+d*x))^4*sqrt(e*(c+d*x))/(d*e)-8*b*Unintegrable((a+b*arcsinh(c+d*x))^3*sqrt(e*(c+d*x))/sqrt(1+(c+d*x)^2),x)/e],
[(a+b*arcsinh(c+d*x))^4/(c*e+d*e*x)^(3/2),x,2,-2*(a+b*arcsinh(c+d*x))^4/(d*e*sqrt(e*(c+d*x)))+8*b*Unintegrable((a+b*arcsinh(c+d*x))^3/(sqrt(e*(c+d*x))*sqrt(1+(c+d*x)^2)),x)/e],
[(a+b*arcsinh(c+d*x))^4/(c*e+d*e*x)^(5/2),x,2,-2/3*(a+b*arcsinh(c+d*x))^4/(d*e*(e*(c+d*x))^(3/2))+8/3*b*Unintegrable((a+b*arcsinh(c+d*x))^3/((e*(c+d*x))^(3/2)*sqrt(1+(c+d*x)^2)),x)/e],
[(a+b*arcsinh(c+d*x))^4/(c*e+d*e*x)^(7/2),x,2,-2/5*(a+b*arcsinh(c+d*x))^4/(d*e*(e*(c+d*x))^(5/2))+8/5*b*Unintegrable((a+b*arcsinh(c+d*x))^3/((e*(c+d*x))^(5/2)*sqrt(1+(c+d*x)^2)),x)/e],

# n<0

# Integrands of the form (1+(a+b x)^2)^(m/2) ArcSinh[a +b x]^n

# m>0
[arcsinh(a+b*x)^3*sqrt(1+a^2+2*a*b*x+b^2*x^2),x,7,-3/8*(a+b*x)^2/b-3/8*arcsinh(a+b*x)^2/b-3/4*(a+b*x)^2*arcsinh(a+b*x)^2/b+1/8*arcsinh(a+b*x)^4/b+3/4*(a+b*x)*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b+1/2*(a+b*x)*arcsinh(a+b*x)^3*sqrt(1+(a+b*x)^2)/b],
[arcsinh(a+b*x)^2*sqrt(1+a^2+2*a*b*x+b^2*x^2),x,6,-1/4*arcsinh(a+b*x)/b-1/2*(a+b*x)^2*arcsinh(a+b*x)/b+1/6*arcsinh(a+b*x)^3/b+1/4*(a+b*x)*sqrt(1+(a+b*x)^2)/b+1/2*(a+b*x)*arcsinh(a+b*x)^2*sqrt(1+(a+b*x)^2)/b],
[arcsinh(a+b*x)*sqrt(1+a^2+2*a*b*x+b^2*x^2),x,4,-1/4*(a+b*x)^2/b+1/4*arcsinh(a+b*x)^2/b+1/2*(a+b*x)*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b],
[sqrt(1+a^2+2*a*b*x+b^2*x^2)/arcsinh(a+b*x),x,5,1/2*Chi(2*arcsinh(a+b*x))/b+1/2*log(arcsinh(a+b*x))/b],
[sqrt(1+a^2+2*a*b*x+b^2*x^2)/arcsinh(a+b*x)^2,x,6,(-1-(a+b*x)^2)/(b*arcsinh(a+b*x))+Shi(2*arcsinh(a+b*x))/b],
[sqrt(1+a^2+2*a*b*x+b^2*x^2)/arcsinh(a+b*x)^3,x,4,1/2*(-1-(a+b*x)^2)/(b*arcsinh(a+b*x)^2)+Chi(2*arcsinh(a+b*x))/b-(a+b*x)*sqrt(1+(a+b*x)^2)/(b*arcsinh(a+b*x))],
[(1+a^2+2*a*b*x+b^2*x^2)^(3/2)*arcsinh(a+b*x)^3,x,15,-51/128*(a+b*x)^2/b-3/128*(a+b*x)^4/b+3/32*(a+b*x)*(1+(a+b*x)^2)^(3/2)*arcsinh(a+b*x)/b-27/128*arcsinh(a+b*x)^2/b-9/16*(a+b*x)^2*arcsinh(a+b*x)^2/b-3/16*(1+(a+b*x)^2)^2*arcsinh(a+b*x)^2/b+1/4*(a+b*x)*(1+(a+b*x)^2)^(3/2)*arcsinh(a+b*x)^3/b+3/32*arcsinh(a+b*x)^4/b+45/64*(a+b*x)*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b+3/8*(a+b*x)*arcsinh(a+b*x)^3*sqrt(1+(a+b*x)^2)/b],
[(1+a^2+2*a*b*x+b^2*x^2)^(3/2)*arcsinh(a+b*x)^2,x,11,1/32*(a+b*x)*(1+(a+b*x)^2)^(3/2)/b-9/64*arcsinh(a+b*x)/b-3/8*(a+b*x)^2*arcsinh(a+b*x)/b-1/8*(1+(a+b*x)^2)^2*arcsinh(a+b*x)/b+1/4*(a+b*x)*(1+(a+b*x)^2)^(3/2)*arcsinh(a+b*x)^2/b+1/8*arcsinh(a+b*x)^3/b+15/64*(a+b*x)*sqrt(1+(a+b*x)^2)/b+3/8*(a+b*x)*arcsinh(a+b*x)^2*sqrt(1+(a+b*x)^2)/b],
[(1+a^2+2*a*b*x+b^2*x^2)^(3/2)*arcsinh(a+b*x),x,7,-5/16*(a+b*x)^2/b-1/16*(a+b*x)^4/b+1/4*(a+b*x)*(1+(a+b*x)^2)^(3/2)*arcsinh(a+b*x)/b+3/16*arcsinh(a+b*x)^2/b+3/8*(a+b*x)*arcsinh(a+b*x)*sqrt(1+(a+b*x)^2)/b],
[(1+a^2+2*a*b*x+b^2*x^2)^(3/2)/arcsinh(a+b*x),x,6,1/2*Chi(2*arcsinh(a+b*x))/b+1/8*Chi(4*arcsinh(a+b*x))/b+3/8*log(arcsinh(a+b*x))/b],
[(1+a^2+2*a*b*x+b^2*x^2)^(3/2)/arcsinh(a+b*x)^2,x,7,-(1+(a+b*x)^2)^2/(b*arcsinh(a+b*x))+Shi(2*arcsinh(a+b*x))/b+1/2*Shi(4*arcsinh(a+b*x))/b],
[(1+a^2+2*a*b*x+b^2*x^2)^(3/2)/arcsinh(a+b*x)^3,x,11,-1/2*(1+(a+b*x)^2)^2/(b*arcsinh(a+b*x)^2)-2*(a+b*x)*(1+(a+b*x)^2)^(3/2)/(b*arcsinh(a+b*x))+Chi(2*arcsinh(a+b*x))/b+Chi(4*arcsinh(a+b*x))/b],

# m<0
[arcsinh(a+b*x)^3/sqrt(1+a^2+2*a*b*x+b^2*x^2),x,2,1/4*arcsinh(a+b*x)^4/b],
[arcsinh(a+b*x)^2/sqrt(1+a^2+2*a*b*x+b^2*x^2),x,2,1/3*arcsinh(a+b*x)^3/b],
[arcsinh(a+b*x)/sqrt(1+a^2+2*a*b*x+b^2*x^2),x,2,1/2*arcsinh(a+b*x)^2/b],
[1/(arcsinh(a+b*x)*sqrt(1+a^2+2*a*b*x+b^2*x^2)),x,2,log(arcsinh(a+b*x))/b],
[1/(arcsinh(a+b*x)^2*sqrt(1+a^2+2*a*b*x+b^2*x^2)),x,2,(-1)/(b*arcsinh(a+b*x))],
[1/(arcsinh(a+b*x)^3*sqrt(1+a^2+2*a*b*x+b^2*x^2)),x,2,(-1/2)/(b*arcsinh(a+b*x)^2)],
[arcsinh(a+b*x)^3/(1+a^2+2*a*b*x+b^2*x^2)^(3/2),x,8,arcsinh(a+b*x)^3/b-3*arcsinh(a+b*x)^2*log(1+exp(2*arcsinh(a+b*x)))/b-3*arcsinh(a+b*x)*polylog(2,-exp(2*arcsinh(a+b*x)))/b+3/2*polylog(3,-exp(2*arcsinh(a+b*x)))/b+(a+b*x)*arcsinh(a+b*x)^3/(b*sqrt(1+(a+b*x)^2))],
[arcsinh(a+b*x)^2/(1+a^2+2*a*b*x+b^2*x^2)^(3/2),x,7,arcsinh(a+b*x)^2/b-2*arcsinh(a+b*x)*log(1+exp(2*arcsinh(a+b*x)))/b-polylog(2,-exp(2*arcsinh(a+b*x)))/b+(a+b*x)*arcsinh(a+b*x)^2/(b*sqrt(1+(a+b*x)^2))],
[arcsinh(a+b*x)/(1+a^2+2*a*b*x+b^2*x^2)^(3/2),x,3,-1/2*log(1+(a+b*x)^2)/b+(a+b*x)*arcsinh(a+b*x)/(b*sqrt(1+(a+b*x)^2))],
[1/((1+a^2+2*a*b*x+b^2*x^2)^(3/2)*arcsinh(a+b*x)),x,1,Unintegrable(1/((1+(a+b*x)^2)^(3/2)*arcsinh(a+b*x)),x)],
[1/((1+a^2+2*a*b*x+b^2*x^2)^(3/2)*arcsinh(a+b*x)^2),x,2,(-1)/(b*(1+(a+b*x)^2)*arcsinh(a+b*x))-2*Unintegrable((a+b*x)/((1+(a+b*x)^2)^2*arcsinh(a+b*x)),x)],

# Integrands of the form u ArcSinh[a x^n]^p

# Integrands of the form x^m ArcSinh[a x^n]

# n>0
[x^3*arcsinh(a*x^2),x,5,1/8*arcsinh(a*x^2)/a^2+1/4*x^4*arcsinh(a*x^2)-1/8*x^2*sqrt(1+a^2*x^4)/a],
[x^2*arcsinh(a*x^2),x,4,1/3*x^3*arcsinh(a*x^2)-2/9*x*sqrt(1+a^2*x^4)/a+1/9*(1+a*x^2)*sqrt(cos(2*arctan(x*sqrt(a)))^2)/cos(2*arctan(x*sqrt(a)))*EllipticF(sin(2*arctan(x*sqrt(a))),sqrt(1/2))*sqrt((1+a^2*x^4)/(1+a*x^2)^2)/(a^(3/2)*sqrt(1+a^2*x^4))],
[x*arcsinh(a*x^2),x,3,1/2*x^2*arcsinh(a*x^2)-1/2*sqrt(1+a^2*x^4)/a],
[arcsinh(a*x^2),x,5,x*arcsinh(a*x^2)-2*x*sqrt(1+a^2*x^4)/(1+a*x^2)+2*(1+a*x^2)*sqrt(cos(2*arctan(x*sqrt(a)))^2)/cos(2*arctan(x*sqrt(a)))*EllipticE(sin(2*arctan(x*sqrt(a))),sqrt(1/2))*sqrt((1+a^2*x^4)/(1+a*x^2)^2)/(sqrt(a)*sqrt(1+a^2*x^4))-(1+a*x^2)*sqrt(cos(2*arctan(x*sqrt(a)))^2)/cos(2*arctan(x*sqrt(a)))*EllipticF(sin(2*arctan(x*sqrt(a))),sqrt(1/2))*sqrt((1+a^2*x^4)/(1+a*x^2)^2)/(sqrt(a)*sqrt(1+a^2*x^4))],
[arcsinh(a*x^2)/x,x,5,-1/4*arcsinh(a*x^2)^2+1/2*arcsinh(a*x^2)*log(1-exp(2*arcsinh(a*x^2)))+1/4*polylog(2,exp(2*arcsinh(a*x^2)))],
[arcsinh(a*x^2)/x^2,x,3,-arcsinh(a*x^2)/x+(1+a*x^2)*sqrt(cos(2*arctan(x*sqrt(a)))^2)/cos(2*arctan(x*sqrt(a)))*EllipticF(sin(2*arctan(x*sqrt(a))),sqrt(1/2))*sqrt(a)*sqrt((1+a^2*x^4)/(1+a*x^2)^2)/sqrt(1+a^2*x^4)],
[arcsinh(a*x^2)/x^3,x,5,-1/2*arcsinh(a*x^2)/x^2-1/2*a*arctanh(sqrt(1+a^2*x^4))],
[arcsinh(a*x^2)/x^4,x,6,-1/3*arcsinh(a*x^2)/x^3-2/3*a*sqrt(1+a^2*x^4)/x+2/3*a^2*x*sqrt(1+a^2*x^4)/(1+a*x^2)-2/3*a^(3/2)*(1+a*x^2)*sqrt(cos(2*arctan(x*sqrt(a)))^2)/cos(2*arctan(x*sqrt(a)))*EllipticE(sin(2*arctan(x*sqrt(a))),sqrt(1/2))*sqrt((1+a^2*x^4)/(1+a*x^2)^2)/sqrt(1+a^2*x^4)+1/3*a^(3/2)*(1+a*x^2)*sqrt(cos(2*arctan(x*sqrt(a)))^2)/cos(2*arctan(x*sqrt(a)))*EllipticF(sin(2*arctan(x*sqrt(a))),sqrt(1/2))*sqrt((1+a^2*x^4)/(1+a*x^2)^2)/sqrt(1+a^2*x^4)],
[arcsinh(a*x^5)/x,x,5,-1/10*arcsinh(a*x^5)^2+1/5*arcsinh(a*x^5)*log(1-exp(2*arcsinh(a*x^5)))+1/10*polylog(2,exp(2*arcsinh(a*x^5)))],
[x^2*arcsinh(sqrt(x)),x,7,5/48*arcsinh(sqrt(x))+1/3*x^3*arcsinh(sqrt(x))+5/72*x^(3/2)*sqrt(1+x)-1/18*x^(5/2)*sqrt(1+x)-5/48*sqrt(x)*sqrt(1+x)],
[x*arcsinh(sqrt(x)),x,6,-3/16*arcsinh(sqrt(x))+1/2*x^2*arcsinh(sqrt(x))-1/8*x^(3/2)*sqrt(1+x)+3/16*sqrt(x)*sqrt(1+x)],
[arcsinh(sqrt(x)),x,6,1/2*arcsinh(sqrt(x))+x*arcsinh(sqrt(x))-1/2*sqrt(x)*sqrt(1+x)],
[arcsinh(sqrt(x))/x,x,5,-arcsinh(sqrt(x))^2+2*arcsinh(sqrt(x))*log(1-exp(2*arcsinh(sqrt(x))))+polylog(2,exp(2*arcsinh(sqrt(x))))],
[arcsinh(sqrt(x))/x^2,x,3,-arcsinh(sqrt(x))/x-sqrt(1+x)/sqrt(x)],
[arcsinh(sqrt(x))/x^3,x,4,-1/2*arcsinh(sqrt(x))/x^2-1/6*sqrt(1+x)/x^(3/2)+1/3*sqrt(1+x)/sqrt(x)],
[arcsinh(sqrt(x))/x^4,x,5,-1/3*arcsinh(sqrt(x))/x^3-1/15*sqrt(1+x)/x^(5/2)+4/45*sqrt(1+x)/x^(3/2)-8/45*sqrt(1+x)/sqrt(x)],
[arcsinh(sqrt(x))/x^5,x,6,-1/4*arcsinh(sqrt(x))/x^4-1/28*sqrt(1+x)/x^(7/2)+3/70*sqrt(1+x)/x^(5/2)-2/35*sqrt(1+x)/x^(3/2)+4/35*sqrt(1+x)/sqrt(x)],

# n<0
[x^2*arcsinh(a/x),x,6,1/3*x^3*arccsch(x/a)-1/6*a^3*arctanh(sqrt(1+a^2/x^2))+1/6*a*x^2*sqrt(1+a^2/x^2)],
[x*arcsinh(a/x),x,3,1/2*x^2*arccsch(x/a)+1/2*a*x*sqrt(1+a^2/x^2)],
[arcsinh(a/x),x,5,x*arccsch(x/a)+a*arctanh(sqrt(1+a^2/x^2))],
[arcsinh(a/x)/x,x,5,1/2*arcsinh(a/x)^2-arcsinh(a/x)*log(1-exp(2*arcsinh(a/x)))-1/2*polylog(2,exp(2*arcsinh(a/x)))],
[arcsinh(a/x)/x^2,x,3,-arccsch(x/a)/x+sqrt(1+a^2/x^2)/a],
[arcsinh(a/x)/x^3,x,5,-1/4*arccsch(x/a)/a^2-1/2*arccsch(x/a)/x^2+1/4*sqrt(1+a^2/x^2)/(a*x)],
[arcsinh(a/x)/x^4,x,5,1/9*(1+a^2/x^2)^(3/2)/a^3-1/3*arccsch(x/a)/x^3-1/3*sqrt(1+a^2/x^2)/a^3],

# Integrands of the form x^m ArcSinh[a x^n] with n symbolic
[x^m*arcsinh(a*x^n),x,3,x^(1+m)*arcsinh(a*x^n)/(1+m)-a*n*x^(1+m+n)*hypergeom([1/2,1/2*(1+m+n)/n],[1/2*(1+m+3*n)/n],-a^2*x^(2*n))/((1+m)*(1+m+n))],
[x^2*arcsinh(a*x^n),x,3,1/3*x^3*arcsinh(a*x^n)-1/3*a*n*x^(3+n)*hypergeom([1/2,1/2*(3+n)/n],[3/2*(1+n)/n],-a^2*x^(2*n))/(3+n)],
[x*arcsinh(a*x^n),x,3,1/2*x^2*arcsinh(a*x^n)-1/2*a*n*x^(2+n)*hypergeom([1/2,1/2*(2+n)/n],[1/2*(3+2/n)],-a^2*x^(2*n))/(2+n)],
[arcsinh(a*x^n),x,3,x*arcsinh(a*x^n)-a*n*x^(1+n)*hypergeom([1/2,1/2*(1+n)/n],[1/2*(3+1/n)],-a^2*x^(2*n))/(1+n)],
[arcsinh(a*x^n)/x,x,5,-1/2*arcsinh(a*x^n)^2/n+arcsinh(a*x^n)*log(1-exp(2*arcsinh(a*x^n)))/n+1/2*polylog(2,exp(2*arcsinh(a*x^n)))/n],
[arcsinh(a*x^n)/x^2,x,3,-arcsinh(a*x^n)/x-a*n*x^(-1+n)*hypergeom([1/2,1/2*(-1+n)/n],[1/2*(3+(-1)/n)],-a^2*x^(2*n))/(1-n)],
[arcsinh(a*x^n)/x^3,x,3,-1/2*arcsinh(a*x^n)/x^2-1/2*a*n*x^(-2+n)*hypergeom([1/2,1/2*(1+(-2)/n)],[1/2*(3+(-2)/n)],-a^2*x^(2*n))/(2-n)],

# Integrands of the form (a+b ArcSinh[c+d x^2])^n

# Integrands of the form (a+b ArcSinh[c+d x^2])^n when c^2=-1
[(a+b*arcsinh(I+d*x^2))^4,x,3,384*b^4*x+48*b^2*x*(a+I*b*arcsin(1-I*d*x^2))^2+x*(a+I*b*arcsin(1-I*d*x^2))^4-192*b^3*(a+I*b*arcsin(1-I*d*x^2))*sqrt(2*I*d*x^2+d^2*x^4)/(d*x)-8*b*(a+I*b*arcsin(1-I*d*x^2))^3*sqrt(2*I*d*x^2+d^2*x^4)/(d*x)],
[(a+b*arcsinh(I+d*x^2))^3,x,5,24*a*b^2*x+24*I*b^3*x*arcsin(1-I*d*x^2)+x*(a+I*b*arcsin(1-I*d*x^2))^3-48*b^3*sqrt(2*I*d*x^2+d^2*x^4)/(d*x)-6*b*(a+I*b*arcsin(1-I*d*x^2))^2*sqrt(2*I*d*x^2+d^2*x^4)/(d*x)],
[(a+b*arcsinh(I+d*x^2))^2,x,2,8*b^2*x+x*(a+I*b*arcsin(1-I*d*x^2))^2-4*b*(a+I*b*arcsin(1-I*d*x^2))*sqrt(2*I*d*x^2+d^2*x^4)/(d*x)],
[a+b*arcsinh(I+d*x^2),x,4,a*x+I*b*x*arcsin(1-I*d*x^2)-2*b*sqrt(2*I*d*x^2+d^2*x^4)/(d*x)],
[1/(a+b*arcsinh(I+d*x^2)),x,1,1/2*x*Ci(-1/2*I*(a+I*b*arcsin(1-I*d*x^2))/b)*(I*cosh(1/2*a/b)-sinh(1/2*a/b))/(b*(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2))))-1/2*x*Si(1/2*I*a/b-1/2*arcsin(1-I*d*x^2))*(I*cosh(1/2*a/b)+sinh(1/2*a/b))/(b*(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2))))],
[1/(a+b*arcsinh(I+d*x^2))^2,x,1,1/4*x*Ci(-1/2*I*(a+I*b*arcsin(1-I*d*x^2))/b)*(cosh(1/2*a/b)-I*sinh(1/2*a/b))/(b^2*(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2))))+1/4*x*Si(1/2*I*a/b-1/2*arcsin(1-I*d*x^2))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))/(b^2*(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2))))-1/2*sqrt(2*I*d*x^2+d^2*x^4)/(b*d*x*(a+I*b*arcsin(1-I*d*x^2)))],
[1/(a+b*arcsinh(I+d*x^2))^3,x,2,-1/8*x/(b^2*(a+I*b*arcsin(1-I*d*x^2)))+1/16*x*Ci(-1/2*I*(a+I*b*arcsin(1-I*d*x^2))/b)*(I*cosh(1/2*a/b)-sinh(1/2*a/b))/(b^3*(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2))))-1/16*x*Si(1/2*I*a/b-1/2*arcsin(1-I*d*x^2))*(I*cosh(1/2*a/b)+sinh(1/2*a/b))/(b^3*(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2))))-1/4*sqrt(2*I*d*x^2+d^2*x^4)/(b*d*x*(a+I*b*arcsin(1-I*d*x^2))^2)],
[(a+b*arcsinh(-I+d*x^2))^4,x,3,384*b^4*x+48*b^2*x*(a-I*b*arcsin(1+I*d*x^2))^2+x*(a-I*b*arcsin(1+I*d*x^2))^4-192*b^3*(a-I*b*arcsin(1+I*d*x^2))*sqrt(-2*I*d*x^2+d^2*x^4)/(d*x)-8*b*(a-I*b*arcsin(1+I*d*x^2))^3*sqrt(-2*I*d*x^2+d^2*x^4)/(d*x)],
[(a+b*arcsinh(-I+d*x^2))^3,x,5,24*a*b^2*x-24*I*b^3*x*arcsin(1+I*d*x^2)+x*(a-I*b*arcsin(1+I*d*x^2))^3-48*b^3*sqrt(-2*I*d*x^2+d^2*x^4)/(d*x)-6*b*(a-I*b*arcsin(1+I*d*x^2))^2*sqrt(-2*I*d*x^2+d^2*x^4)/(d*x)],
[(a+b*arcsinh(-I+d*x^2))^2,x,2,8*b^2*x+x*(a-I*b*arcsin(1+I*d*x^2))^2-4*b*(a-I*b*arcsin(1+I*d*x^2))*sqrt(-2*I*d*x^2+d^2*x^4)/(d*x)],
[a+b*arcsinh(-I+d*x^2),x,4,a*x-I*b*x*arcsin(1+I*d*x^2)-2*b*sqrt(-2*I*d*x^2+d^2*x^4)/(d*x)],
[1/(a+b*arcsinh(-I+d*x^2)),x,1,1/2*x*Shi(1/2*(a-I*b*arcsin(1+I*d*x^2))/b)*(cosh(1/2*a/b)+I*sinh(1/2*a/b))/(b*(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2))))-1/2*x*Ci(1/2*I*(a-I*b*arcsin(1+I*d*x^2))/b)*(I*cosh(1/2*a/b)+sinh(1/2*a/b))/(b*(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2))))],
[1/(a+b*arcsinh(-I+d*x^2))^2,x,1,1/4*x*Ci(1/2*I*(a-I*b*arcsin(1+I*d*x^2))/b)*(cosh(1/2*a/b)+I*sinh(1/2*a/b))/(b^2*(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2))))-1/4*x*Shi(1/2*(a-I*b*arcsin(1+I*d*x^2))/b)*(I*cosh(1/2*a/b)+sinh(1/2*a/b))/(b^2*(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2))))-1/2*sqrt(-2*I*d*x^2+d^2*x^4)/(b*d*x*(a-I*b*arcsin(1+I*d*x^2)))],
[1/(a+b*arcsinh(-I+d*x^2))^3,x,2,-1/8*x/(b^2*(a-I*b*arcsin(1+I*d*x^2)))+1/16*x*Shi(1/2*(a-I*b*arcsin(1+I*d*x^2))/b)*(cosh(1/2*a/b)+I*sinh(1/2*a/b))/(b^3*(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2))))-1/16*x*Ci(1/2*I*(a-I*b*arcsin(1+I*d*x^2))/b)*(I*cosh(1/2*a/b)+sinh(1/2*a/b))/(b^3*(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2))))-1/4*sqrt(-2*I*d*x^2+d^2*x^4)/(b*d*x*(a-I*b*arcsin(1+I*d*x^2))^2)],

# Integrands of the form (a+b ArcSinh[c+d x^2])^(n/2) when c^2=-1
[(a+b*arcsinh(I+d*x^2))^(5/2),x,2,x*(a+I*b*arcsin(1-I*d*x^2))^(5/2)+15*b^2*x*FresnelS(sqrt((-I)/b)*sqrt(a+I*b*arcsin(1-I*d*x^2))/sqrt(Pi))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2)))*sqrt((-I)/b))-15*b^3*x*FresnelC(sqrt((-I)/b)*sqrt(a+I*b*arcsin(1-I*d*x^2))/sqrt(Pi))*(I*cosh(1/2*a/b)+sinh(1/2*a/b))*sqrt((-I)/b)*sqrt(Pi)/(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2)))-5*b*(a+I*b*arcsin(1-I*d*x^2))^(3/2)*sqrt(2*I*d*x^2+d^2*x^4)/(d*x)+15*b^2*x*sqrt(a+I*b*arcsin(1-I*d*x^2))],
[(a+b*arcsinh(I+d*x^2))^(3/2),x,2,x*(a+I*b*arcsin(1-I*d*x^2))^(3/2)-3*b^2*x*FresnelS(sqrt(a+I*b*arcsin(1-I*d*x^2))/(sqrt(I*b)*sqrt(Pi)))*(cosh(1/2*a/b)-I*sinh(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2)))*sqrt(I*b))+3*b*x*FresnelC(sqrt(a+I*b*arcsin(1-I*d*x^2))/(sqrt(I*b)*sqrt(Pi)))*(I*cosh(1/2*a/b)-sinh(1/2*a/b))*sqrt(I*b)*sqrt(Pi)/(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2)))-3*b*sqrt(2*I*d*x^2+d^2*x^4)*sqrt(a+I*b*arcsin(1-I*d*x^2))/(d*x)],
[(a+b*arcsinh(I+d*x^2))^(1/2),x,1,x*FresnelS(sqrt((-I)/b)*sqrt(a+I*b*arcsin(1-I*d*x^2))/sqrt(Pi))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2)))*sqrt((-I)/b))-b*x*FresnelC(sqrt((-I)/b)*sqrt(a+I*b*arcsin(1-I*d*x^2))/sqrt(Pi))*(I*cosh(1/2*a/b)+sinh(1/2*a/b))*sqrt((-I)/b)*sqrt(Pi)/(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2)))+x*sqrt(a+I*b*arcsin(1-I*d*x^2))],
[1/(a+b*arcsinh(I+d*x^2))^(1/2),x,1,-x*FresnelS(sqrt(a+I*b*arcsin(1-I*d*x^2))/(sqrt(I*b)*sqrt(Pi)))*(cosh(1/2*a/b)-I*sinh(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2)))*sqrt(I*b))-x*FresnelC(sqrt(a+I*b*arcsin(1-I*d*x^2))/(sqrt(I*b)*sqrt(Pi)))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2)))*sqrt(I*b))],
[1/(a+b*arcsinh(I+d*x^2))^(3/2),x,1,-((-I)/b)^(3/2)*x*FresnelC(sqrt((-I)/b)*sqrt(a+I*b*arcsin(1-I*d*x^2))/sqrt(Pi))*(cosh(1/2*a/b)-I*sinh(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2)))+((-I)/b)^(3/2)*x*FresnelS(sqrt((-I)/b)*sqrt(a+I*b*arcsin(1-I*d*x^2))/sqrt(Pi))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2)))-sqrt(2*I*d*x^2+d^2*x^4)/(b*d*x*sqrt(a+I*b*arcsin(1-I*d*x^2)))],
[1/(a+b*arcsinh(I+d*x^2))^(5/2),x,2,-1/3*x*FresnelS(sqrt(a+I*b*arcsin(1-I*d*x^2))/(sqrt(I*b)*sqrt(Pi)))*(cosh(1/2*a/b)-I*sinh(1/2*a/b))*sqrt(Pi)/(b^2*(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2)))*sqrt(I*b))-1/3*x*FresnelC(sqrt(a+I*b*arcsin(1-I*d*x^2))/(sqrt(I*b)*sqrt(Pi)))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))*sqrt(Pi)/(b^2*(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2)))*sqrt(I*b))-1/3*sqrt(2*I*d*x^2+d^2*x^4)/(b*d*x*(a+I*b*arcsin(1-I*d*x^2))^(3/2))-1/3*x/(b^2*sqrt(a+I*b*arcsin(1-I*d*x^2)))],
[1/(a+b*arcsinh(I+d*x^2))^(7/2),x,2,-1/15*x/(b^2*(a+I*b*arcsin(1-I*d*x^2))^(3/2))-1/15*((-I)/b)^(3/2)*x*FresnelC(sqrt((-I)/b)*sqrt(a+I*b*arcsin(1-I*d*x^2))/sqrt(Pi))*(cosh(1/2*a/b)-I*sinh(1/2*a/b))*sqrt(Pi)/(b^2*(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2))))+1/15*((-I)/b)^(3/2)*x*FresnelS(sqrt((-I)/b)*sqrt(a+I*b*arcsin(1-I*d*x^2))/sqrt(Pi))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))*sqrt(Pi)/(b^2*(cos(1/2*arcsin(1-I*d*x^2))-sin(1/2*arcsin(1-I*d*x^2))))-1/5*sqrt(2*I*d*x^2+d^2*x^4)/(b*d*x*(a+I*b*arcsin(1-I*d*x^2))^(5/2))-1/15*sqrt(2*I*d*x^2+d^2*x^4)/(b^3*d*x*sqrt(a+I*b*arcsin(1-I*d*x^2)))],
[(a+b*arcsinh(-I+d*x^2))^(5/2),x,2,x*(a-I*b*arcsin(1+I*d*x^2))^(5/2)+15*b^2*x*FresnelS(sqrt(I/b)*sqrt(a-I*b*arcsin(1+I*d*x^2))/sqrt(Pi))*(cosh(1/2*a/b)-I*sinh(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))*sqrt(I/b))-15*b^2*x*FresnelC(sqrt(I/b)*sqrt(a-I*b*arcsin(1+I*d*x^2))/sqrt(Pi))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))*sqrt(I/b))-5*b*(a-I*b*arcsin(1+I*d*x^2))^(3/2)*sqrt(-2*I*d*x^2+d^2*x^4)/(d*x)+15*b^2*x*sqrt(a-I*b*arcsin(1+I*d*x^2))],
[(a+b*arcsinh(-I+d*x^2))^(3/2),x,2,x*(a-I*b*arcsin(1+I*d*x^2))^(3/2)-3*b^2*x*FresnelS(sqrt(a-I*b*arcsin(1+I*d*x^2))/(sqrt(-I*b)*sqrt(Pi)))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))*sqrt(-I*b))-3*b*x*FresnelC(sqrt(a-I*b*arcsin(1+I*d*x^2))/(sqrt(-I*b)*sqrt(Pi)))*(I*cosh(1/2*a/b)+sinh(1/2*a/b))*sqrt(-I*b)*sqrt(Pi)/(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))-3*b*sqrt(-2*I*d*x^2+d^2*x^4)*sqrt(a-I*b*arcsin(1+I*d*x^2))/(d*x)],
[(a+b*arcsinh(-I+d*x^2))^(1/2),x,1,x*FresnelS(sqrt(I/b)*sqrt(a-I*b*arcsin(1+I*d*x^2))/sqrt(Pi))*(cosh(1/2*a/b)-I*sinh(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))*sqrt(I/b))-x*FresnelC(sqrt(I/b)*sqrt(a-I*b*arcsin(1+I*d*x^2))/sqrt(Pi))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))*sqrt(I/b))+x*sqrt(a-I*b*arcsin(1+I*d*x^2))],
[1/(a+b*arcsinh(-I+d*x^2))^(1/2),x,1,-x*FresnelC(sqrt(a-I*b*arcsin(1+I*d*x^2))/(sqrt(-I*b)*sqrt(Pi)))*(cosh(1/2*a/b)-I*sinh(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))*sqrt(-I*b))-x*FresnelS(sqrt(a-I*b*arcsin(1+I*d*x^2))/(sqrt(-I*b)*sqrt(Pi)))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))*sqrt(-I*b))],
[1/(a+b*arcsinh(-I+d*x^2))^(3/2),x,1,(I/b)^(3/2)*x*FresnelS(sqrt(I/b)*sqrt(a-I*b*arcsin(1+I*d*x^2))/sqrt(Pi))*(cosh(1/2*a/b)-I*sinh(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))-(I/b)^(3/2)*x*FresnelC(sqrt(I/b)*sqrt(a-I*b*arcsin(1+I*d*x^2))/sqrt(Pi))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))-sqrt(-2*I*d*x^2+d^2*x^4)/(b*d*x*sqrt(a-I*b*arcsin(1+I*d*x^2)))],
[1/(a+b*arcsinh(-I+d*x^2))^(5/2),x,2,-1/3*x*FresnelS(sqrt(a-I*b*arcsin(1+I*d*x^2))/(sqrt(-I*b)*sqrt(Pi)))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))*sqrt(Pi)/(b^2*(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2)))*sqrt(-I*b))-1/3*x*FresnelC(sqrt(a-I*b*arcsin(1+I*d*x^2))/(sqrt(-I*b)*sqrt(Pi)))*(I*cosh(1/2*a/b)+sinh(1/2*a/b))*sqrt(-I*b)*sqrt(Pi)/(b^3*(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2))))-1/3*sqrt(-2*I*d*x^2+d^2*x^4)/(b*d*x*(a-I*b*arcsin(1+I*d*x^2))^(3/2))-1/3*x/(b^2*sqrt(a-I*b*arcsin(1+I*d*x^2)))],
[1/(a+b*arcsinh(-I+d*x^2))^(7/2),x,2,-1/15*x/(b^2*(a-I*b*arcsin(1+I*d*x^2))^(3/2))-1/15*(I/b)^(3/2)*x*FresnelC(sqrt(I/b)*sqrt(a-I*b*arcsin(1+I*d*x^2))/sqrt(Pi))*(cosh(1/2*a/b)+I*sinh(1/2*a/b))*sqrt(Pi)/(b^2*(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2))))+1/15*x*FresnelS(sqrt(I/b)*sqrt(a-I*b*arcsin(1+I*d*x^2))/sqrt(Pi))*(I*cosh(1/2*a/b)+sinh(1/2*a/b))*sqrt(I/b)*sqrt(Pi)/(b^3*(cos(1/2*arcsin(1+I*d*x^2))-sin(1/2*arcsin(1+I*d*x^2))))-1/5*sqrt(-2*I*d*x^2+d^2*x^4)/(b*d*x*(a-I*b*arcsin(1+I*d*x^2))^(5/2))-1/15*sqrt(-2*I*d*x^2+d^2*x^4)/(b^3*d*x*sqrt(a-I*b*arcsin(1+I*d*x^2)))],

# Integrands of the form u^m (a+b ArcSinh[Sqrt[1-c x]/Sqrt[1+c x]])^n
[(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))^n/(1-c^2*x^2),x,0,Unintegrable((a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))^n/(1-c^2*x^2),x)],
[(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))^3/(1-c^2*x^2),x,8,-1/4*(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))^4/(b*c)-(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))^3*log(1+(-1)/exp(2*arcsinh(sqrt(1-c*x)/sqrt(1+c*x))))/c+3/2*b*(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))^2*polylog(2,exp(-2*arcsinh(sqrt(1-c*x)/sqrt(1+c*x))))/c+3/2*b^2*(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))*polylog(3,exp(-2*arcsinh(sqrt(1-c*x)/sqrt(1+c*x))))/c+3/4*b^3*polylog(4,exp(-2*arcsinh(sqrt(1-c*x)/sqrt(1+c*x))))/c],
[(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))^2/(1-c^2*x^2),x,7,-1/3*(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))^3/(b*c)-(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))^2*log(1+(-1)/exp(2*arcsinh(sqrt(1-c*x)/sqrt(1+c*x))))/c+b*(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))*polylog(2,exp(-2*arcsinh(sqrt(1-c*x)/sqrt(1+c*x))))/c+1/2*b^2*polylog(3,exp(-2*arcsinh(sqrt(1-c*x)/sqrt(1+c*x))))/c],
[(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))/(1-c^2*x^2),x,6,-1/2*(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))^2/(b*c)-(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))*log(1+(-1)/exp(2*arcsinh(sqrt(1-c*x)/sqrt(1+c*x))))/c+1/2*b*polylog(2,exp(-2*arcsinh(sqrt(1-c*x)/sqrt(1+c*x))))/c],
[1/((1-c^2*x^2)*(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))),x,0,Unintegrable(1/((1-c^2*x^2)*(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))),x)],
[1/((1-c^2*x^2)*(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))^2),x,0,Unintegrable(1/((1-c^2*x^2)*(a+b*arcsinh(sqrt(1-c*x)/sqrt(1+c*x)))^2),x)],

# Integrands involving inverse hyperbolic sines of exponentials

# x^m ArcSinh[c E^(a+b x)]
[arcsinh(c*exp(a+b*x)),x,6,-1/2*arcsinh(c*exp(a+b*x))^2/b+arcsinh(c*exp(a+b*x))*log(1-exp(2*arcsinh(c*exp(a+b*x))))/b+1/2*polylog(2,exp(2*arcsinh(c*exp(a+b*x))))/b],

# Integrands involving exponentials of inverse hyperbolic sines

# x^m E^(ArcSinh[a+b x]^n)
[exp(arcsinh(a+b*x))*x^3,x,5,1/48/(b^4*exp(3*arcsinh(a+b*x)))+3/16*a/(b^4*exp(2*arcsinh(a+b*x)))+1/8*(-1+6*a^2)/(b^4*exp(arcsinh(a+b*x)))+1/16*a*(3-4*a^2)*exp(2*arcsinh(a+b*x))/b^4-1/24*(1-6*a^2)*exp(3*arcsinh(a+b*x))/b^4-3/32*a*exp(4*arcsinh(a+b*x))/b^4+1/80*exp(5*arcsinh(a+b*x))/b^4+1/8*a*(3-4*a^2)*arcsinh(a+b*x)/b^4],
[exp(arcsinh(a+b*x))*x^2,x,5,(-1/16)/(b^3*exp(2*arcsinh(a+b*x)))-1/2*a/(b^3*exp(arcsinh(a+b*x)))-1/16*(1-4*a^2)*exp(2*arcsinh(a+b*x))/b^3-1/6*a*exp(3*arcsinh(a+b*x))/b^3+1/32*exp(4*arcsinh(a+b*x))/b^3-1/8*(1-4*a^2)*arcsinh(a+b*x)/b^3],
[exp(arcsinh(a+b*x))*x,x,5,1/4/(b^2*exp(arcsinh(a+b*x)))-1/4*a*exp(2*arcsinh(a+b*x))/b^2+1/12*exp(3*arcsinh(a+b*x))/b^2-1/2*a*arcsinh(a+b*x)/b^2],
[exp(arcsinh(a+b*x)),x,5,1/4*exp(2*arcsinh(a+b*x))/b+1/2*arcsinh(a+b*x)/b],
[exp(arcsinh(a+b*x))/x,x,9,b*x+a*arcsinh(a+b*x)+a*log(x)-arctanh((1+a^2+a*b*x)/(sqrt(1+a^2)*sqrt(1+a^2+2*a*b*x+b^2*x^2)))*sqrt(1+a^2)+sqrt(1+a^2+2*a*b*x+b^2*x^2)],
[exp(arcsinh(a+b*x))/x^2,x,9,-a/x+b*arcsinh(a+b*x)+b*log(x)-a*b*arctanh((1+a^2+a*b*x)/(sqrt(1+a^2)*sqrt(1+a^2+2*a*b*x+b^2*x^2)))/sqrt(1+a^2)-sqrt(1+a^2+2*a*b*x+b^2*x^2)/x],
[exp(arcsinh(a+b*x))/x^3,x,6,-1/2*a/x^2-b/x-1/2*b^2*arctanh((1+a^2+a*b*x)/(sqrt(1+a^2)*sqrt(1+a^2+2*a*b*x+b^2*x^2)))/(1+a^2)^(3/2)-1/2*(1+a^2+a*b*x)*sqrt(1+a^2+2*a*b*x+b^2*x^2)/((1+a^2)*x^2)],
[exp(arcsinh(a+b*x))/x^4,x,7,-1/3*a/x^3-1/2*b/x^2-1/3*(1+a^2+2*a*b*x+b^2*x^2)^(3/2)/((1+a^2)*x^3)+1/2*a*b^3*arctanh((1+a^2+a*b*x)/(sqrt(1+a^2)*sqrt(1+a^2+2*a*b*x+b^2*x^2)))/(1+a^2)^(5/2)+1/2*a*b*(1+a^2+a*b*x)*sqrt(1+a^2+2*a*b*x+b^2*x^2)/((1+a^2)^2*x^2)],
[exp(arcsinh(a+b*x))/x^5,x,8,-1/4*a/x^4-1/3*b/x^3-1/4*(1+a^2+2*a*b*x+b^2*x^2)^(3/2)/((1+a^2)*x^4)+5/12*a*b*(1+a^2+2*a*b*x+b^2*x^2)^(3/2)/((1+a^2)^2*x^3)+1/8*(1-4*a^2)*b^4*arctanh((1+a^2+a*b*x)/(sqrt(1+a^2)*sqrt(1+a^2+2*a*b*x+b^2*x^2)))/(1+a^2)^(7/2)+1/8*(1-4*a^2)*b^2*(1+a^2+a*b*x)*sqrt(1+a^2+2*a*b*x+b^2*x^2)/((1+a^2)^3*x^2)],
[exp(arcsinh(a+b*x)^2)*x^3,x,37,-1/32*erfi(-2+arcsinh(a+b*x))*sqrt(Pi)/(b^4*exp(4))+1/16*erfi(-1+arcsinh(a+b*x))*sqrt(Pi)/(b^4*E)-3/8*a^2*erfi(-1+arcsinh(a+b*x))*sqrt(Pi)/(b^4*E)-1/16*erfi(1+arcsinh(a+b*x))*sqrt(Pi)/(b^4*E)+3/8*a^2*erfi(1+arcsinh(a+b*x))*sqrt(Pi)/(b^4*E)+1/32*erfi(2+arcsinh(a+b*x))*sqrt(Pi)/(b^4*exp(4))-3/16*a*erfi(1/2*(-3+2*arcsinh(a+b*x)))*sqrt(Pi)/(b^4*exp(9/4))+3/16*a*erfi(1/2*(-1+2*arcsinh(a+b*x)))*sqrt(Pi)/(b^4*exp(1/4))-1/4*a^3*erfi(1/2*(-1+2*arcsinh(a+b*x)))*sqrt(Pi)/(b^4*exp(1/4))+3/16*a*erfi(1/2*(1+2*arcsinh(a+b*x)))*sqrt(Pi)/(b^4*exp(1/4))-1/4*a^3*erfi(1/2*(1+2*arcsinh(a+b*x)))*sqrt(Pi)/(b^4*exp(1/4))-3/16*a*erfi(1/2*(3+2*arcsinh(a+b*x)))*sqrt(Pi)/(b^4*exp(9/4))],
[exp(arcsinh(a+b*x)^2)*x^2,x,27,1/4*a*erfi(-1+arcsinh(a+b*x))*sqrt(Pi)/(b^3*E)-1/4*a*erfi(1+arcsinh(a+b*x))*sqrt(Pi)/(b^3*E)+1/16*erfi(1/2*(-3+2*arcsinh(a+b*x)))*sqrt(Pi)/(b^3*exp(9/4))-1/16*erfi(1/2*(-1+2*arcsinh(a+b*x)))*sqrt(Pi)/(b^3*exp(1/4))+1/4*a^2*erfi(1/2*(-1+2*arcsinh(a+b*x)))*sqrt(Pi)/(b^3*exp(1/4))-1/16*erfi(1/2*(1+2*arcsinh(a+b*x)))*sqrt(Pi)/(b^3*exp(1/4))+1/4*a^2*erfi(1/2*(1+2*arcsinh(a+b*x)))*sqrt(Pi)/(b^3*exp(1/4))+1/16*erfi(1/2*(3+2*arcsinh(a+b*x)))*sqrt(Pi)/(b^3*exp(9/4))],
[exp(arcsinh(a+b*x)^2)*x,x,17,-1/8*erfi(-1+arcsinh(a+b*x))*sqrt(Pi)/(b^2*E)+1/8*erfi(1+arcsinh(a+b*x))*sqrt(Pi)/(b^2*E)-1/4*a*erfi(1/2*(-1+2*arcsinh(a+b*x)))*sqrt(Pi)/(b^2*exp(1/4))-1/4*a*erfi(1/2*(1+2*arcsinh(a+b*x)))*sqrt(Pi)/(b^2*exp(1/4))],
[exp(arcsinh(a+b*x)^2),x,7,1/4*erfi(1/2*(-1+2*arcsinh(a+b*x)))*sqrt(Pi)/(b*exp(1/4))+1/4*erfi(1/2*(1+2*arcsinh(a+b*x)))*sqrt(Pi)/(b*exp(1/4))],
[exp(arcsinh(a+b*x)^2)/x,x,0,CannotIntegrate(exp(arcsinh(a+b*x)^2)/x,x)],
[exp(arcsinh(a+b*x)^2)/x^2,x,0,CannotIntegrate(exp(arcsinh(a+b*x)^2)/x^2,x)],

# Miscellaneous integrands involving inverse hyperbolic sines
[arcsinh(a+b*x)/(a*d/b+d*x),x,7,-1/2*arcsinh(a+b*x)^2/d+arcsinh(a+b*x)*log(1-exp(2*arcsinh(a+b*x)))/d+1/2*polylog(2,exp(2*arcsinh(a+b*x)))/d],
[x/(arcsinh(x)*sqrt(1+x^2)),x,2,Shi(arcsinh(x))],
[x^3*arcsinh(a+b*x^4),x,4,1/4*(a+b*x^4)*arcsinh(a+b*x^4)/b-1/4*sqrt(1+(a+b*x^4)^2)/b],
[x^(-1+n)*arcsinh(a+b*x^n),x,4,(a+b*x^n)*arcsinh(a+b*x^n)/(b*n)-sqrt(1+(a+b*x^n)^2)/(b*n)],
[arcsinh(c/(a+b*x)),x,6,(a+b*x)*arccsch(a/c+b*x/c)/b+c*arctanh(sqrt(1+1/(a/c+b*x/c)^2))/b],
[x/arcsinh(sinh(x)),x,-1,arcsinh(sinh(x))+log(arcsinh(sinh(x)))*(-arcsinh(sinh(x))+x*sech(x)*sqrt(cosh(x)^2))],
[arcsinh(sqrt(-1+b*x^2))^n/sqrt(-1+b*x^2),x,2,arcsinh(sqrt(-1+b*x^2))^(1+n)*sqrt(b*x^2)/(b*(1+n)*x)],
[1/(arcsinh(sqrt(-1+b*x^2))*sqrt(-1+b*x^2)),x,2,log(arcsinh(sqrt(-1+b*x^2)))*sqrt(b*x^2)/(b*x)]]:
